QA    !>11    W4/W    1«^H   ■*■» 
UNIVERSITY   Of   CALIFORNIA    SAN   Di^^^^^^ 


3   1822  01062  3031 


r- 


LIBRARY 

UNIVERSITY  OP 
CALIFORNIA 
SAN   DI£SQ 


tl4   511   W478   18<)«  ; 


3  1822  01062  3031  ^A   - 

SSE 


MATHEMATICAL    TEXT-BOOKS 

BY 

GEORGE  A.  WENTWORTH 


Mental  Arithmetic 

Elementary  Arithmetic 

Practical  Arithmetic 

Primary  Arithmetic    (Wentworth  and  Reed) 

Grammar  School  Arithmetic 

Advanced  Arithmetic 

First  Steps  in  Algebra 

New  School  Algebra 

School  Algebra 

Elements  of  Algebra  (Revised  Edition) 

Shorter  Course  in  Algebra 

Complete  Algebra 

Higher  Algebra 

College  Algebra  (Revised  Edition) 

First  Steps  in  Geometry    (Wentworth  and  Hill) 

Plane  Geometry  (Revised) 

Solid  Geometry  (Revised) 

Plane  and  Solid  Geometry  (Revised) 

Syllabus  of  Geometry 

Geometrical  Exercises 

Analytic  Geometry 

Plane  and  Solid  Geometry  and  Plane  Trigonometry 

(Second  Revised  Edition i 
Plane  Trigonometry  (Second  Revised  Edition) 
Plane    Trigonometry    and    Tables    (Second    Revised 

Edition) 
Plane  and  Spherical  Trigonometry  (Second  Revised 

Edition) 
Plane  and   Spherical   Trigonometry  and    Tables 

(Second  Revised  Edition) 
Plane  and   Spherical   Trigonometry,  Surveying,  and 

Navigation  (Second  Revised  Edition) 
Surveying  and  Tables  (Second  Revised  Edition) 
Plane  Trigonometry,  Surveying,  and  Tables  (Second 

Revised  Editioni 
Plane  and   Spherical  Trigonometry,  Surveying,  and 

Tables  (Second  Revised  Edition) 
Logarithms,  Metric  Measures,  etc. 


ELEMENTS 


Analytic    Geometry 


G.  A.  WENTWUKTH,  A.M. 

ACTHOE  OF  A  SKKIKS  OK  TEXT-BOOKS  IN  MATHEMATICS 


GINN  &  COMPANY 

BOSTON  ■  NEW  YORK  •  CHICAGO  •  LONDON 


Copyright,  1886,  by 
GEORGE  A.  WENTWORTH 


ALL  RIGHTS  RESERVED 
G8.10 


Che   gtftengum   ^regg 

GINN   &   COMPANY  •  PRO- 
FRIETORS  •  BOSTON  •  U.S.A. 


NOTE   TO   THE    SECOND   EDITION. 

In  this  edition  such  changes  have  been  made  as  actual  experi- 
ence in  the  class-room  has  shown  to  be  desirable. 

A  chapter  on  Higher  Plane  Curves,  and  four  chapters  on  Solid 
Geometry  have  been  added,  making  the  work  sufficiently  extensive 
for  our  best  schools  and  colleges. 

An  effort  has  been  made  to  have  this  edition  free  from  errors. 
It  is  not  likely,  however,  that  this  effort  has  been  entirely  success- 
ful, and  the  author  will  be  very  grateful  to  any  reader  who  will 
notify  him  of  any  needed  corrections. 


NOTE   TO   THE   EDITION   OF   1898. 

The  old  plates  have  become  so  worn  that  it  is  necessary  to  have 
new  plates.  A  few  verbal  changes  liave  been  made.  No  changes, 
however,  have  been  made  that  will  prevent  the  using  of  old  and 
new  books  together. 

March,  1898.  G.  A  WENTWORTH. 


PREFACE. 


o>Ko 


rpmS  book  is  intended  for  beginners.  As  beginners 
-*~  generally  find  great  diiticulty  in  comprehending  the 
connection  between  a  locus  and  its  equation,  the  opening 
chapter  is  devoted  maiidy  to  an  attempt,  by  means  of  easy 
illustrations  and  exam[)les,  to  niaki;  this  connection  clear. 

Each  chapter  abounds  in  exercises  ;  for  it  is  only  by 
solving  })roblems  which  require  some  degree  of  original 
thought  that  any  real  mastery  of  the  study  can  be  gained. 

The  more  difficult  propositions  have  been  })ut  at  the  ends 
of  the  chapters,  under  the  heading  of  "  Supplementary 
Propositions."  This  arrangement  makes  it  possible  for 
every  teacher  to  mark  out  his  own  course.  The  simplest 
course  will  be  Chapters  I. -ill.  and  Chapters  V.-VII.,  with 
Review  Exercises  and  Supplementary  Propositions  left 
out.  Between  this  course  and  the  entire  work  the  teacher 
can  exercise  his  choice,  and  take  just  so  much  as  time  and 
circumstances  will  allow. 

The  author  has  gathered  his  materials  from  many  sources, 
but  he  is  particularly  indebted  to  tlie  English  treatise  of 
Charles  Smith.  Special  acknowledgment  is  due  to  ii.  A. 
Hill,  A.M.,  of  Cambridge,  Mass.,  and  to  Prof.  J.  M. 
Taylor,  Colgate  University,  Hamilton,  New  York,  for 
assistance  in  the  preparation  of  the  book. 

Corrections  and  suggestions  will  be  thankfully  received. 

G.  A.  WENTWORTH. 
Exeter,  N.H.,  January,  1888. 


O  O  IsT  T  E  N  T  S. 


ol»<c 


PART   I.     PLANE   GEOMETEY. 


Chapter  I.     Loci  and  their  Equations. 

BECTION 

1.  Quadrants      ..... 

2.  Algebraic  Signs      .... 

3.  Axes  of  Coordinates 

4.  Rectilinear  System  of  Coordinates 

5.  Circular  Measure  of  an  Angle 
6-7.  Distance  between  Two  Points 
8-9.  Division  of  a  Line  . 

10-16.  Constants  and  Variables 

17-24.  Locus  of  an  Equation     . 

25.  Definitions 

26.  Intercepts  of  a  Curve 

27.  Intersections  of  Two  Curves 

28.  Curve  Passing  through  the  Origin  . 

29.  Equation  having  no  Constant  Term 

30.  Construction  of  Straight  Line  and  Circle 
31-34.  Constructions  of  Loci  of  Given  E<iuations 

35.  Equation  of  a  Curve       .... 

Review  Exercises 


page 

1 

2 
2 
3 
5 
6 
8 

10 
14 
22 
22 
22 
23 
23 
25 
25 
31 
33 


Chapter  II.     The  Straight  Line. 


36.  Notation 

37-39.  Equations  of  the  Straight  Line 

40.  Symmetrical  Equation  of  the  Straight  Line 

41.  Normal  Equation  of  the  Straight  Line    . 
42-43.  General  Equation  of  the  First  Degree     . 


36 
36 
38 
39 
43 


VI 


CONTENTS. 


SECTION 

44. 
45. 
46. 

47. 

48-49. 

50. 


Locus  of  the  First  Order        ..... 
Angle  Formed  by  Two  Lines  .... 

Equati(Mis  of  Parallels  and  Perpendiculars     . 
Equation  of  Line  making  given  Angle  with  a  Line 
Distance  from  a  Point  to  a  Line     .... 

Area  of  a  Triangle 

Review  Exercises 


PAGE 

43 
45 
46 
46 

50 

54 
M 


Supplementary  Propositions. 

51-52.  Equation  of  a  Line  from  a  Point  to  tlie  Intersection  of 

Two  Lines 

53.  Condition  that  Three  Lines  meet  in  a  Point  . 

54.  Equation  of  the  Bisector  of  an  Angle     . 

55.  Homogeneous  Equation  of  the  nlh  Degree     . 

56.  Angles  between  the  Two  Lines  Ax-  +  Cxy  +  By^  =  0 

57.  Condition  that  a  Quadratic  represents  Two  Straight  Lines 

58.  Problems  on  Loci  involving  Three  Variables. 


Chapter  III.     The  Circle. 

59-00.  Equations  of  the  Circle 

61.  Condition  that  a  Quadratic  represents  a  Circle 

62.  Condition  that  a  Point  is  without,  on,  or  within  a 

63.  Tangents,  Normals,  Subtangents,  Subnormals 

64.  Equation  of  a  Tangent  to  the  Circle  x-  +  ij-  —  r^ 
(55.  P^quation  of  a  Normal  through  the  Point  (xi,  ii\) 

66.  Equations  of   the  Tangent  and  Normal  to  the 

(X  —  a)-  +  (?/  —  6)2  =  r2  . 

67.  Condition  that  a  Straight  Line  touches  a  Circle 
Review  Exercises 


Circle 


Circle 


Supplementary  Propositions. 

68.  Diameter,  Chords  of  a  Diameter    .... 

69.  Equation  of  a  Diameter  of  the  Circle  x^  +  y-  =  r^ 

70.  Condition  of  Two,  One,  or  No  Tangents  to  a  Circle 

71.  Equation  of  the  Chord  of  Contact  .... 

72.  Pole  and  Polar.     Equation  of  Polar 

73.  Pole  and  Polar  of  a  Circle      ..... 


CONTENTS. 


SECTIOK  PAOE 


74.  Relations  of  Poles  and  Polars 

75.  Geometrical  Construction  of  a  Polar  to  a  Circle 
70.  Length  of  Tangent  from  a  given  Point  . 

77.  Radical  Axis  of  Two  Circles  .... 

78.  Radical  Centre  of  Two  Circles 


Chapter  IV.     Different  Systems  of  Coordinates. 

79-81.  Rectilinear  and  Oblique  Systems    . 

82.  Polar  System  .... 

83.  Polar  Equation  of  the  Circle  . 

84.  Transformation  of  Coordinates 

85.  New  Axes  parallel  to  Old  Axes 

86.  From  One  Set  of  Rectangular  Axes  to  Another  Set 

87.  From  One  Set   of  Rectangular   Axes  to  Another  Set 

with  Different  ( )rigin 

88.  From  Rectangular  to  ( )blique  Axes 

89.  From  Rectangular  to  Polar  Coordinates 

90.  From  Polar  to  Rectangular  Coiirdinates 

91.  Degree  of  an  Equation  not  Altered  by  Transformation 
Review  Exercises 


Chapter  V.     The  Parabola. 

92.  Simple  Proprieties  of  the  Parabola 

93.  Construction  of  a  Parabola 

94.  Principal  Equation  of  the  Parabola 

95.  Parabola  Symmetrical  with  Respect  to  the  Axis    . 

96.  Condition   that  a  I'oint   is   without,   on,   or   within 

Parabola  ....... 

97.  Latus  Rectum  a  Third  Proportional  to  any   Abscissa 

and  Corresponding  Ordinate  .... 

98.  Squares  of  Ordinates  of  Two  Points  are  as  Abscissas 

99.  Points  in  which  a  Straight  Line  meets  a  Parabola. 

100.  Equations  of  Tangents  and  Normals 

101.  Subtangent  and  Subnormal 

102.  Tangent  makes  E(iual  Angles  with  the  Axis  and  Focal 

Radius 

Review  Exercises 


!):5 
94 
05 
05 
06 


99 
101 
103 
105 
105 
100 

107 
107 
108 
100 
109 
111 


113 
113 
114 
115 

115 

lie. 

lit! 
110 

no 
no 

1-20 
123 


VUl  CONTEXTS. 

SCPPLEMENTARV    PhOPOSITIONS. 
SECTION  PAGE 

103.  Condition  of  Two,  One,  or  No  Tangents  to  a  Parabola  .  126 

104.  Equation  of  tlie  Cliord  of  Contact 127 

105.  Equation  of  the  Polar  with  respect  to  the  Parabola        .  127 

106.  Equation  of  a  Diameter  of  the  Parabola         .         .         .  128 

107.  Tangent  through  End  of  a  Diameter  Parallel  to  Chords 

of  Diameter    ........  129 

108.  Perpendicular  from  Focus  to  a  Chord,  also  from  Focus 

to  a  Tangent 129 

109.  Tangents  through  the  Ends  of  a  Chord .         .         .         .130 

110.  Locus  of  Foot  of  Perpendicular  from  Focus  to  a  Tangent  130 

111.  Points  from  which  each  Point  in  Tangent  is  Equidistant  130 

112.  Tangents  at  Right  Angles  intersect  in  Directrix     .         .  130 

113.  Polar  of  the  Focus 131 

114.  Equation  of  the  Parabola,  Axes  being  Diameter  and 

Tangent  through  its  End 131 

115.  Polar  Equation  of  the  Parabola 133 

Chapter  VI.     The  Ellipse. 

116.  Simple  Properties  of  the  Ellipse 136 

117.  Construction  of  an  Ellipse 136 

118.  Transverse  and  Conjugate  Axes 137 

119.  Equation  of  the  Ellipse 138 

120.  Characteristics  of  the  Curve  learned  from  its  Equation  139 

121.  Change  in  the  Form  of  the  Ellipse  by  Changing  Semi- 

Axes       139 

122.  Ratio  of  the  Squares  of  Any  Two  Ordinates  .        .         .  139 

123.  Condition  that  a  Point  is  without,  on,  or  within  the 

Ellipse 140 

124.  Form  of  Equation  representing  an  Ellipse      .        .         .  140 

125.  Latus  Rectum  a  Third  Proportional  to  Major  and  Minor 

Axes       .........  141 

126.  Auxiliary  Circles 141 

127.  Ratio  of  the  Ordinates  of  the  Ellipse  and  Auxiliary  Circle  142 

128.  Construction  of  the  Ellipse  by  §  127       .         .         .         .  142 

129.  Area  of  the  Ellipse 143 

130.  Equations  of  Tangents  and  Normals       ....  146 

131.  Subtangents  and  Subnormals          .        *.         .         .         .  147 

132.  Tangents  to  Ellipses  having  a  Common  Major  Axis       .  148 


CONTENTS. 


SECTION 
133. 

134. 

135. 
136. 


137. 
138. 
139. 
140. 
141. 
142. 
143. 

144. 

145. 
146. 

147. 
148. 
149. 
150. 
151. 


152. 
153. 
154. 
155. 
156. 
157. 
158. 
159. 
160. 
161. 
162. 
163. 


The  Normal  bisects  Angle  between  Focal  Radii     . 
Method  of  drawing  the  Tangent  and  Normal  at  a  Point 

on  Ellipse 
Equation  of  Tangent  in  Terms  of  its  Slope 
Director  Circle  of  the  Ellipse 
Review  Exercises  . 


Supplementary  Propositions. 

Condition  of  Two,  One,  or  No  Tangents  to  an  Ellipse 

Equation  of  Chord  of  Contact        .... 

Equation  of  the  Polar  with  respect  to  an  Ellipse    . 

Method  of  drawing  a  Tangent  to  an  Ellipse  . 

Equation  of  a  Diameter  of  an  Ellipse     . 

Conjugate  Diameters 

Tangents  at  Ends  of  Diameter  Parallel  to  Conjugate 
Diameter 

Relation  of  Ends  of  Conjugate  Diameters 

Sum  of  Squares  of  any  Pair  of  Semi-Conjugate  Diameters 

Difference  between  Eccenti'ic  Angles  of  Ends  of  Conju- 
gate Diameters 

Angle  between  Two  Conjugate  Diameters 

Conjugate  Diameters  parallel  to  Supplemental  Chords  . 

Equation  of  Ellipse  having  Conjugate  Diameters  as  Axes 

Construction  of  the  Polar  of  a  Focus     .... 

The  Polar  Equation  with  the  Left-Hand  Focus  as  Pole 

Chapter  VII,     The  Hyperbola. 

Simple  Properties  of  the  Hyperbola 
Construction  of  an  Hyperbola 
Centre,  Transverse  Axis,  Vertices 
Equation  of  the  Hyperbola 
Properties  of  the  Hyperbola 
Equilateral  Hyperbola  . 
Conjugate  Hyperbolas    . 
Straight  Line  through  Centre  meets  Cur 
Asymptotes    . 
Equation  of  Tangent 
Equation  of  Normal 
Subtangent,  Subnormal 


PAGE 

148 

149 
140 
150 
152 


154 
154 
155 
155 
156 
156 


e  in  Two  Points 


157 
158 
158 

158 
159 
160 
161 
162 
163 


168 
168 
170 
171 
171 
172 
172 
173 
173 
175 
175 
175 


X  CONTKNTS. 

SECTIOK  PAGE 

l(i4.     Cniiditiiin  that  a  Straiglit  Line  is  Tangent      .         .         .  175 

1()5.     iMiuation  of  the  Director  Circle      .         .         .         .         .175 

IGO.     Tangent  and  Normal  bisect  Angles  between  Focal  Kadii  175 

Review  Exercises 177 

SUPPLEMICNTAUY    Puoi'OSITIONS. 

107.     Condition  of  Two,  One,  or  No  Tangents  to  an  Hyperbola  178 
10b.     K(iuation  of  Chord  of  Contact        .         .         .         .         .178 

16!).     Ivjuation  of  the  Polar  with  respect  to  the  Hyperbola     .  179 

170.  Etiuation  of  a  Diameter  of  an  Hyjierbola       .         .         .  17!) 

171.  Conjugate  Diameters 179 

172.  Properties  of  Conjugate  Diameters         ....  179 
178.     Length  of  a  Diameter     .......  180 

174.  Portions  of  a  Line  between  two  Conjugate  Hyperbolas 

are  Ecjual ISO 

175.  Tangent  at  End  of  a  Diameter  is  Parallel  to  Conjugate 

Diameter         .         .         .         .         .         .         .         .180 

176.  Given  Eml   of   Diameter,  to  find   Ends  of   Conjugate 

Diameter 181 

177.  Equation  of  Hyperbola  having  Conjugate  Diameters  as 

Axes 182 

178.  Tangents   at   Ends   of   Conjugate    Diameters   meet  in 

Asymptotes 182 

179.  Angle  between  Two  Conjugate  Diameters      .         .         .  183 

180.  P(jrtions  of  a  Line  between  Hyperbola  and  Asymptotes 

are  Eiiual 18.3 

181.  A  Parallel  to  an  Asymptote  meets  the  Curve  in  only 

One  Finite  Point 184 

182.  Equation  of  Hyperbola  having  the  Asynqjtotes  as  Axes  185 

183.  The  Polar  of  the  Focus 187 

184.  I'olar  Equations  of  an  Hyperbola 188 


Chapter  VIII.     Loci  of  the  Second  Order. 

185.  General  Equation  of  the  second  Degree 

186.  Condition  that  this  Equation  represents  Two  Lines 

187.  Central  and  Non-Central  Curves    .... 

188.  General  E(iuation  of  Central  Loci  .... 
18!).  lieduction  of  this  Equation  to  a  Known  Form 
190.  Nature  of  Locus  of  Px-  +  Qy-  =  R 


191 
191 
192 
192 
194 
195 


CONTENTS. 


SECTION 

191. 
192. 
193. 
194. 
195. 
196. 


197. 
198. 
199. 
200. 
201. 
202-203. 
204. 
205. 
206. 
207. 
208. 
209. 


Locus  of  Equation  when  A  =  0  and  2  =  0. 
Locus  of  Equation  when  A  is  not  0  and  S  =  0 
Summary    ....... 

Examples    ....... 

Definition  of  a  Conic  ..... 

Equation  of  a  Conic    ..... 

Exercises    . 


Chapter  IX.     IIic.nER  Plane  Curves. 

Higher  Plane  Curves  . 
The  Cissoid  of  Diodes 
The  Conchoid  of  Niconicdes 
The  Lemniscate  of  Bernnulli 
The  Witch  of  Agnesi  . 
The  Cycloid 

Spirals         .... 
The  Spiral  of  Archimedes  . 
The  Hyperbolic  Spiral 
The  Lituus. 
The  Logarithmic  Spiral 
The  Parabolic  Spiral  . 


PAGE 

196 
197 
200 
201 
205 
205 
206 


208 
208 
211 
213 
215 
216 
220 
221 
222 
223 
223 
224 


210 
211 
212-213 
214-216 
217 
218 
219 
220 


PART   II.     SOLID   GEOMETRY. 

Chapter  I.     The  Point. 

Definitions ...... 

The  Radius  Vector  of  a  Point     . 
Direction  Angles  and  Direction  Cosines 
Projections  upon  a  Straight  Line 
Angle  Between  Two  Straight  Lines    . 
Distance  Between  Two  Points    . 
Polar  Coordinates       .... 

Projections  upon  a  Plane    . 
Exercises 


226 
228 
228 
230 
231 
232 
233 
234 
235 


Chapter  II.     The  Plane. 
221-222.     Normal  Equation  of  a  Plane 


236 


CONTENTS. 


SECTION 

223.  Sjninietrical  Equation  of  a  Plane 

224.  Angle  between  Two  Planes 

225.  Distance  from  a  Point  to  a  Plane 


226. 

227-228. 

229. 

230. 


231. 

2.32. 

233. 

234-235. 


236. 
237. 
238. 
239. 
240. 
241. 
242. 
243. 
244. 
245. 


Chapter  III.     The  Straight  Line 

Equations  of  a  Straight  Line 
Symmetrical  Equations  of  a  Straight  Line 
Angle  between  Two  Straight  Lines    . 
Inclination  of  a  Line  to  a  Plane . 
Exercises 


Supplementary  Propositions. 

Traces  of  a  Plane 

Equations  of  the  Traces  of  a  Plane     . 
Condition  of  Intersection  of  Two  Straight  Lines 
To  pass  a  Plane  through  a  Point  and  a  Right  Line 

Chapter  IV.     Surfaces  of  Revolution. 

A  Single  Equation  in  a;,  ?/,  z,  represents  a  Surface 

Ti'aces  of  a  Surface     . 

Definitions .... 

General  Equation  of  a  Surface  of  Revolution 

Paraboloid  of  Revolution    . 

Ellipsoid  of  Revolution 

Hyperboloid  of  Revolution . 

Central  Surfaces 

Cone  of  Revolution     . 

Conic  Sections    . 

Exercises    .... 


page 

238 
239 
240 


243 
245 
246 
246 
247 


250 
250 
250 
251 


252 
254 
254 
254 
255 
256 
258 
259 
259 
260 
263 


Supplementary  Propositions. 

246.  General  Equation  of  the  Sphere 

247.  Intersection  of  Two  Spheres 

248.  Equation  of  Tangent  Plane  to  Sphere 
249-250.  Transformation  of  Coordinates  . 

251-252.  Quadrics 

253-257.  Central  Quadrics         .... 

258.  Non-Central  Quadrics 


264 
265 
265 
266 
267 
268 
271 


ANALYTIC    GEOMETRY. 


3j«<0« 


PART  I.  — PLANE   GEOMETRY. 


CHAPTER  I. 
LOCI   AND   THEIR  EQUATIONS. 

Rectilinear  System  of  Coordinates. 

1.  Let  AX' and  FF'  (Fig.  1)  be  two  fixed  lines  inter- 
secting in  the  point  0.  These  lines  divide  the  plane  in 
which  they  lie  into  four  portions. 


■-D 

Y 

<B 

E 

i^ 

\Pi  ^ 

X' 

'F 

0 

M      X 

JPa 

T 

\l^ 

G 

iC 

Y' 

^-H 

Fig.  1. 

Let  these  parts  be  called  Quadrants  (as  in  Trigonometry), 
and  distinguished  by  naming  the  area  between  OX  and 
OY  the  first  quadrant;  that  between  OY  and  OX'  the 
second  quadrant;  that  between  OX'  and  Oi'^  the  third 
quadrant ;  and  that  between  OY  and  OX  the  fourth 
quadrant. 


ANALYTIC    GEOMETKY. 


Sui)])Ose  the  position  of  a  point  is  described  by  saying 
that  its  distance  from  YY',  expressed  in  terms  of  some 
chosen  unit  of  length,  is  3,  and  its  distance  from  XX'  is  4, 
it  being  understood  that  tlie  distance  from  either  line  is 
measured  parallel  to  the  other.  It  is  clear  that  in  each 
quadrant  there  is  one  point,  and  only  one,  that  will 
satisfy  these  conditions.  The  position  of  the  point  in 
each  quadrant  may  be  found  by  drawing  parallels  to  YY' 
at  the  distance  3  from  YY',  and  parallels  to  XX  at  the 
distance  4  fro.m  XX' ;  then  the  intersections  JPi,  P^,  Ps, 
Pi  satisfy  the  given  conditions. 


D 

Y 

,B 

E 

[^ 

\Pi  -f 

X' 

jiV 

0 

M\      X 

iPs 

T 

.._  \3 

a 

iC 

Y' 

\ah 

Fig.  1. 

2.  In  order  to  determine  which  one  of  the  four  points, 
Pi,  Pg,  Ps,  Pi,  is  meant,  we  adopt  the  rule  that  opposite 
directions  shall  be  indicated  by  V7illke  signs.  As  in  Trigo- 
nometry, distances  measured  from  YY'  to  the  right  are 
considered  positive ;  to  the  left,  negative.  Distances  meas- 
ured from  XX'  upward  are  positive  ;  doivnward,  negative. 
Then  the  position  of  Pi  will  be  denoted  by  +  3,  +  4 ;  of 
P2,by-3,+4;  of  P3,  by -3, -4  ;    ofP„by  +  3,-4. 

3.  The  fixed  lines  XX'  and.  YY'  are  called  the  Axes  of 
Coordinates ;  XX  is  called  the  Axis  of  Abscissas,  or  Axis 
of  X ;  YY',  the  Axis  of  Ordinates,  or  Axis  of  y.  The 
intersection  0  is  called  the  Origin. 


LOCI    AND    THEIR    EQUATIONS.  3 

The  two  distances  (with  signs  prefixed)  that  deter- 
mine the  position  of  a  point  are  called  the  Coordinates 
of  the  point ;  the  distance  of  the  point  from  YY'  is  called 
its  Abscissa ;  and  the  distance  from  XX',  its  Ordinate. 

Abscissas  are  usually  denoted  by  x,  and  ordinates  by  y. 
A  point  is  represented  algebraically  by  simply  writing 
the  values  of  its  coordinates  within  a  parenthesis,  that  of 
the  abscissa  being  always  written  first. 

Thus  Pi  (Fig.  1)  is  the  point  (3,  4) ;  P^,  the  point  ( —  3,  4) ; 
Ps,  the  point  (—3,  —4);  P^,  the  point  (3,-4).  In  general, 
the  point  whose  coordinates  are  x  and  y  is  the  point  (x,  y). 

4.  This  method  of  determining  the  position  of  a  point 
in  a  plane  is  called  the  Rectilinear  System  of  Coordinates. 
The  coordinates  are  called  rectangular  or  oblique,  accoi'ding 
as  the  axes  are  rectangular  or  oblique  ;  that  is,  according 
as  the  axes  intersect  at  right  or  oblique  angles.  In  the 
first  three  chapters  we  shall  use  only  rectangular  coordinates. 

Note.  The  first  man  to  employ  this  method  successfully  in 
investigating  the  properties  of  certain  figures  was  the  French  philoso- 
pher Descartes,  whose  work  on  Geometry  appeared  in  the  year  1637. 

Exercise  1. 

1.  What  are  the  coordinates  of  the  origin  ? 

2.  In  what  quadrants  are  the  following  points  (a  and  b 
being  given  lengths)  : 

{-a,-b),    (-a,b),    (a,b),    (a,-b)? 

3.  To  what  quadrants  is  a  point  limited  if  its  abscissa  is 
positive  ?  negative  ?  ordinate  positive  ?  ordinate  negative  ? 

4.  In  what  line  does  a  point  lie  if  its  abscissa  =  0  ?  if 
its  ordinate  ^  0  ? 

5.  A  point  (x,  y)  moves  parallel  to  the  axis  of  x ;  which 
one  of  its  coordinates  remains  constant  in  value  ? 


4  ANALYTIC    GEOMETRY. 

6.  Construct  or ^j^onhe  points  :  (2,3),  (3,  — 3),  (—1,-3), 
(-4,  4),  (3,  0),  (-3,  0),  (0,  4),  (0,  -1),  (0,  0). 

Note.  To  plot  a  point  is  to  mark  its  proper  position  on  paper, 
when  its  coordinates  are  given.  The  fii'st  thing  to  do  is  to  draw  the 
two  axes.     The  rest  of  the  work  is  obvious  after  a  study  of  Nos.  1-3. 

7.  Construct  the  triangle  whose  vertices  are  the  points 
(2,4),  (-2,7),  (-6,-8). 

8.  Construct  the  quadrilateral  whose  vertices  are  the 
points  (7,  2),  (0,  -9),  (-7,  -1),  (-6,  4). 

9.  Construct  the  quadrilateral  whose  vertices  are  ( — 3, 6), 
(—3,  0),  (3,  0),  (3,  6).      What  kind  of  quadrilateral  is  it? 

10.  Mark  the  four  points  (2,  1),  (4,  3),  (2,  5),  and  (0,  3), 
and  connect  them  by  straight  lines.  What  kind  of  a  figure 
do  these  four  lines  enclose  ? 

11.  The  side  of  a  square  =  a  ;  the  origin  of  coordinates 
is  the  intersection  of  the  diagonals.  What  are  the  coor- 
dinates of  the  vertices  (i)  if  the  axes  are  parallel  to  the 
sides  of  the  square  ?  (ii)  if  the  axes  coincide  with  the 
diagonals  ? 

(i.)(»V2.o),(o.«V5),(-|V2,«),(o,-^V5). 

12.  The  side  of  an  equilateral  triangle  =  a  ;  the  origin 
is  taken  at  one  vertex,  and  the  axis  of  x  coincides  with 
one  side.     What  are  the  coordinates  of  the  three  vertices  ? 

Ans.   (0,0),  (a,  0),(^  |^y3). 

13.  The  line  joining  two  points  is  bisected  at  the  origin. 
If  the  coordinates  of  one  of  the  points  are  a  and  b,  what 
are  the  coordinates  of  the  other  ? 

14.  Connect  the  points  (5,  3)  and  (5,  — 3)  by  a  straight 
line.     What  is  the  direction  of  this  line  ? 


liOCI    AND    THE  IK    EQUATIONS.  O 

Circular  Measure. 

5.  In  Analytic  Geometry,  angles  are  often  expressed  in 
degrees,  minutes,  and  seconds ;  but  sometimes  it  is  very 
convenient  to  employ  the  Circular  Measure  of  an  angle. 

In  circular  measure,  an  angle  is  defined  by  the  equation 

,  arc 

angle  =  — - — , 
radius 

in  which  the  word  '•  arc "  denotes  the  length  of  the  arc 

corresponding  to  the  angle  when  both  arc  and  radius  are 

expressed  in  terms  of  a  common  linear  unit. 

This  equation  gives  us  a  correct  measure  of  angular 
magnitude,  because  (as  shown  in  Geometry)  for  a  given 
angle  the  value  of  the  above  ratio  of  arc  and  radius  is 
constant  for  all  values  of  the  radius. 

If  the  radius  =  1,  the  equation  becomes 
angle  =^  arc  ;  that  is. 

In  circular  measure  an  angle  is  measured  hy  the  length  of 
the  arc  suhte^ided  bij  it  iri  a  unit  circle. 

It  is  shown  in  Geometry  that  the  circumference  of  a 
unit  circle  ^  27r  ;  as  this  circumference  contains  360° 
common  measure,  the  two  measures  are  easily  compared 
by  means  of  the  relation 

360  degrees  =  27r  units,  circular  measure. 

Exercise  2. 

1.  Find  the  value  in  circular  measure  of  the  angles  1°, 
45°,  90°,  180°,  270°. 

"*"^-   180'   4'   2'   ''•   2* 

2.  In  circular  measure,  the  unit  angle  is  that  angle 
whose  arc  is  equal  to  the  radius  of  the  circle.  What  is 
the  value  of  this  angle  in  degrees,  etc.? 

Ans   57°  17' 46". 


ANALYTIC    GEOMETRY. 


Distance  between  Two  Points. 

6,    To  find  the  distance  between  two  given  points. 

Let  P  and  Q  (Fig.  2)  be  the  given  points,  cci  and  yi  the 
coordinates  of  P,  ccj  and  y2  those  of  Q.  Also  let  d  =  PQ 
■=■  the  required  distance. 


M 


Fig.  2. 


A^X 


Draw  PJ/and  ^.V  ||  to  OT,  and  PP  ||  to  OX. 
Then  031  =  x„  MP  =  y,, 

ON  =  x^,  NQ  =  ?/2, 

PP  :=  Xa  —  Xi  QR  =  y^  —  y^. 

By  Geometry, 

(^2  =  {x^  —  x^}^  +  (?/2  —  yiY ; 
whence,  rf=  V{x2-iCiy-^+ (1/2- X/i)^- 


[1] 


Since  (xi  —  .rg)^  =  (ir2  —  a"i)^  it  makes  no  difference  which 
point  is  called  (x-^,  y^)  and  which  (x^,  y^). 

7.  Equation  [1]  is  perfectly  general,  holding  true  for 
points  situated  in  any  quadrant.  Thus,  if  P  is  in  the 
second  quadrant  and  Q  in  the  third  quadrant  (Fig.  3), 
^2  —  ^1  is  obviously  equal  to  the  leg  RQ  ;  and  since  y^  is 
negative,  y^  —  yi  is  the  sum  of  two  negative  numbers,  and 
is  equal  to  the  absolute  length  of  the  leg  RP  with  the  — 
sign  prefixed. 


LOCI    AND    THEIR    EQUATIOXS.  7 

Note.  The  learner  should  satisfy  himself  that  equation  [1]  is  per- 
fectly general,  by  constructing  other  special  cases  in  which  tlie  points 
P  and  Q  are  in  different  quadrants.  In  every  case  he  will  find  that 
the  numerical  values  of  the  expressions  {X2  —  Xi)  and  (1/2  —  2/1)  are 
the  legs  of  the  right  triangle,  the  hypotenuse  of  which  is  the  re(iuired 
distance  PQ. 

Equation  [1]  is  merely  an  illustration  of  the  general  truth  that 
theorems  and  formulas  deduced  by  reasoning  loitk  points  or  lines  in  the 
first  quadrant  {where  the  coordinates  are  always  positive)  must,  from 
the  very  nature  of  the  analytic  method,  hold  true  when  the  points  or 
lines  are  situated  in  the  other  quadrants. 

Exercise  3. 

Find  tlie  distance 
>    1.    From  the  point  (—2,  5)  to  the  point  (—8, —3). 

2.  From  the  point  (1,  3)  to  the  point  (6,  15). 

3.  From  the  point  (—4,  5)  to  the  point  (0,  2). 
■^  4.    From  the  origin  to  the  point  ( —  6,  —  8). 

5.  From  the  point  (a,  b)  to  the  point  ( — a,  — b). 

Find  the  lengths  of  the  sides  of  a  triangle 

6.  If  the  vertices  are  the  points  (15,  — 4),  (—9,  3) 
(11,  24). 

V  7.    If  the  verticesare  the  points(2,3),(4,— 5),(— 3,  — 6). 

8.  If  the  vertices  are  the  points  (0,  0),  (3,  4),  (—  3,  4). 

9.  If  the  vertices  are  the  points  (0, 0),  (—a,  0),  (0,  —b). 
"J  10.  The  vertices  of  a  quadrilateral  are  (5,  2),  (3,  7), 
(—1,  4),  (—3,  — 2).  Find  the  lengths  of  the  sides  and 
also  of  the  diagonals. 

V  11.  One  end  of  a  line  whose  length  is  13  is  the  point 
( — 4,  8);  the  ordinate  of  the  other  end  is  3.  What  is  its 
abscissa? 

12.  What  equation  must  the  coordinates  of  the  point 
(x,  7/)  satisfy  if  its  distance  from  the  point  (7,  —  2)  is 
equal  to  11  ? 


8 


ANALYTIC    GEOMETRY. 


13.  What  equation  expresses  algebraically  the  fact  that 
the  point  (x,  y)  is  equidistant  from  the  points  (2,  3)  and 
(4,  5)  ? 

14.  If  the  value  of  a  quantity  depends  on  the  square  of 
a  length,  it  is  immaterial  whether  the  length  is  considered 
positive  or  negative.     Why  ? 

Division  of  a  Line. 
8.    To  bisect  the  line  joining  two  given  2^oints. 

Let  P  and  Q  (Fig.  4)  be  the  given  points  (x^,  i/i)  and 
(^2)  3/2)-  Let  X  and  y  be  the  coordinates  of  H,  the  mid- 
point of  FQ. 

The  meaning  of  the  problem  is  to  find  the  values  of  x 
and  y  in  terms  of  .Tj,  2/1,  and  x^,  y^. 


M      b 


Fig.  4. 


N 


Draw  P3I,  ES,  QJVW  to  OY;  also  draw  PA,  EB  ||  to  OX. 

Then  rt.  A  PEA  =  rt.  A  EQB  (hypotenuse  and  one 
acute  angle  equal). 

Therefore,  PA  =  EB,  and  AE  =  BQ; 

also,  MS=SN. 


By  substitution,    ic  —  a-^  =  .To  —  x,  and  y—y\^yi  —  y\ 
whence,  ^^S^.^^mpi.,  ,-2] 


LOCI    AND    THEIR   EQUATIONS.  M 

9.  To  divide  the  line  joining  two  given  points  into  two 
parts  having  a  given  ratio  m  :  n. 

Let  P  and  Q  (Fig.  5)  be  the  given  points  (x^  y-^  and 
{x^y^.  Let  R  be  the  required  point,  such  that  PR  :  RQ  = 
m  :  n,  and  let  x  and  y  denote  the  coordinates  of  R. 

Complete  the  figure  by  drawing  lines  as  in  Fig.  4. 

The  rt.  A  PR  A  and  RQB,  being  mutually  equiangular, 
are  similar ;  therefore 

PA      PR     m         ,    AR       PR      m 

-= — ,  and 


RB      RQ      n'  BQ       RQ      n 

Substituting  for  the  lines  their  values,  we  have 

X — Xi      m  ,   1/ — 7/,      m 
=— ,    and  ■       ■    — 


1^  Vi—V     *i 

Solving  these  equations  for  x  and  ?/,  we  obtain 

_  ^nx<i  +  ndC\ .      _  tnyi.  +  ny\  [3] 

tn-\-  n    *  tn  +  n 

It  m  =  n,  we  have  the  special  case  of  bisecting  a  line 
already  considered  ;  and  it  is  easy  to  see  that  the  values 
of  X  and  y  reduce  to  the  forms  given  in  [2], 

Exercise  4. 

What  are  the  coordinates  of  the  point 
^1.    Halfway  between  the  points  (5,  3)  and  (7,  9)  ? 

2.    Halfway  between  the  points  (—  6,  2)  and  (4,  —  2)  ? 

-  3.    Halfway  between  the  points  (5,  0)  and  (—1,  —4)  ? 

4.    The    vertices    of    a    triangle    are    (2, 3),    (4,  —  5), 

(—  3,  —  6) ;  find  the  middle  points  of  its  sides. 

T  5.    The  middle  point  of  a  line  is  (6,  4),  and  one  end  of 

.  the  line  is  (5,  7).  What  are  the  coordinates  of  the  other  end  ? 

^  6.    A  line  is  bisected  at  the  origin  ;    one  end  of  the  line 

is  the  point  (—  a,  h).       What  are  the  coordinates  at  the 

other  end  ? 


10  ANALYTIC    GEOMETRY. 

"  7.  Prove  that  the  middle  point  of  the  hypotenuse  of  a 
right  triangle  is  equidistant  from  the  three  vertices. 

8.  Prove  that  the  diagonals  of  a  parallelogram  mutually 
bisect  each  other. 

9.  Show  that  the  values  of  x  and  y  in  [2]  hold  true 
when  the  two  given  points  both  lie  in  the  second  quadrant. 

10.  Solve  the  problem  of  §  9  when  the  line  PQ  is  cut 
externally  instead  of  internally,  in  the  ratio  m  :  n. 

11.  What  are  the  coordinates  of  the  point  that  divides 
the  line  joining  (3,  —  1)  and  (10,  6)  in  the  ratio  3:4? 

12.  The  line  joining  (2,  3)  and  (4,  — 5)  is  trisected. 
Determine  the  point  of  trisection  nearer  (2,  3). 

13.  A  line  AB  is  produced  to  a  point  C,  such  tliat  BC  = 
^  AB.  If  A  and  B  are  the  points  (5,  6)  and  (7,  2),  what 
are  the  coordinates  of  C  ? 

■  14.  A  line  AB  is  produced  to  a  point  C,  such  that  AB  : 
BC  =  4=  :7.  li  A  and  B  are  the  points  (5,  4)  and(6,  —9), 
what  are  the  coordinates  of  C  ? 

15.  Three  vertices  of  a  parallelogram  are  (1,2),  ( — 5,  — 3), 
(7,  —  6).     What  is  the  fourth  vertex? 

Constants  and  Variables. 

10.  In  Analytic  Geometry  a  line  is  regarded  as  a 
geometric  viagnitude  traced  or  generated  by  a  moving  point, 
—  just  as  we  trace  on  paper  what  serves  to  represent  a  line 
to  the  eye  by  moving  the  point  of  a  pen  or  pencil  over  the 
paper. 

We  shall  find  that  great  advantages  are  to  be  gained  by 
defining  a  line  iji  this  way,  but  we  must  be  prepared  from 
the  outset  to  make  an  important  distinction  in  the  use  of. 
symbols  representing  lengths.  We  must  distinguish  between 
symbols  which  denote  definite  or  fixed  lengths  and  those 
which  denote  variable  lengths. 


LOCI    AND    THEIR    EQUATIONS. 


11 


11.  Let  A  (Fig.  6)  be  the  point  (3,  4).  Then  OA  = 
V  9  + 16  =  5.  Now  let  a  jjoint  F  describe  the  line  OA  by 
moving  from  0  to  A,  and  let  the  coordinates  of  Pbe  denoted 
by  X  and  y;  also  let  z  denote  the  length  OP  at  any  position 
of  P.     Then  z  will  increase  continuously  from  0  to  5. 


F 

V 

/ 

^ 

0 

JT 

1 

i      X 

Fig.  6. 

Here  the  word  continuously  deserves  special  attention. 
It  means  that  P  must  pass  successively  through  every 
position  on  the  line  OA  from  0  to  ^  ;  that,  therefore,  z 
must  have  in  succession  every  conceivable  value  between  0 
and  5.  There  will  be  one  position  of  P  for  which  z  is 
equal  to  2 ;  there  will  be  another  position  of  P  for  which  z 
is  equal  to  2.000001;  but  before  reaching  this  value  z  must 
first  pass  through  all  values  between  2  and  2.000001. 

In  the  same  way  x  and  y,  the  coordinates  of  P,  both 
pass  through  a  continuous  change,  x  increasing  continuously 
from  0  to  3,  and  y  from  0  to  4. 

We  may  now  divide  the  lengths  considered  in  this 
example  into  two  classes  : 

(1)  Lengths  supposed  to  remain  constant  in  value, 
namely,  the  coordinates  of  A  and  the  distance  OA  ;  (2) 
lengths  supposed  to  vary  continuously  in  value,  namely, 
the  coordinates  of  P  {x  and  y),  and  the  distance  OP,  or  z. 


12  ANALYTIC    GEOMETRY. 

Quantities  of  the  first  kind  in  any  problem  are  called 
constant  quantities,  or,  more  briefly,  Constants. 

(Quantities  of  the  second  kind  are  called  variable  quanti 
ties,  or,  more  briefly,  Variables. 

12.  Two  variables  are  often  so  related  that  if  one  oi 
them  changes  in  value  the  other  also  changes  in  value. 
The  second  variable  is  then  said  to  be  a  function  of  the 
first  variable.  The  second  variable  is  also  called  the 
dependent  variable,  while  the  first  is  called  the  indej^endent 
variable.  Usually  the  relation  between  two  variables  is 
such  that  either  may  be  treated  as  the  independent  variable, 
and  the  other  as  the  dependent  variable. 

Thus,  in  §  11,  if  we  suppose  z  to  change,  then  both  x  and 
y  will  change  ;  the  values  of  x audi/ then  will  depend  upon 
the  value  given  to  z ;  that  is,  x  and  y  will  be  functions 
of  z.  But  we  may  also  suppose  the  value  of  x,  the  abscissa 
of  P,  to  change  ;  then  it  is  clear  that  the  values  of  both 
y  and  z  must  also  change.  In  this  case  we  take  x  as  the 
independent  variable,  and  values  of  y  and  z  will  depend 
upon  the  value  of  x ;  that  is,  y  and  z  will  be  functions  of  x. 

13.  The  most  concise  way  to  express  the  relations  of 
the  constants  and  variables  which  enter  into  a  problem  is 
by  means  of  algebraic  equations. 

The  coordinates  of  P  (Fig.  6)  throughout  its  motion  are 
always  x  and  y  ;  and  the  triangle  OFM  is  similar  to  the 
triangle  OAB.     Hence,  for  any  position  of  P, 

y-  =  %  and  z^  =  x'^y\ 

By  solving,  2/  "=  q  ^j  ^^^d  ~  =  o  ^. 

o  o 

These  equations  express  the  values  of  y  and  z,  respectively, 

in  terms  of  x  as  the  independent  variable. 


LOCI    AND    THEIR    EQUATIONS.  13 

14.  In  §  11,  instead  of  assuming  3  and  4  as  the  coordi- 
nates of  A,  we  might  have  employed  two  letters,  as  a  and 
b,  with  the  understanding  that  these  letters  should  denote 
two  coordinates  that  remain  constant  in  value  during  the 
motion  of  P.     If  we  choose  these  letters,  we  obtain, 


b  -Ja^  +  b^ 

11=  -X.      z=^ X. 

^      a    '  a 

15.  There  is  a  noteworthy  difference  between  the  con- 
stants 3  and  4  and  the  constants  a  and  h.  The  numbers 
3  and  4  cannot  be  supposed  to  change  under  any  circum- 
stances. The  numbers  a  and  b  are  constants  in  this  sense 
only,  that  they  do  not  change  in  value  when  we  suppose  x 
OT  y  ov  z  to  change  in  value  ;  in  other  words,  they  are  not 
functions  of  x  or  y  or  z  in  the  particular  problem  under 
discussion.  In  all  other  respects  they  are  free  to  represent 
as  many  different  values  as  we  choose  to  assign  to  them. 

Constants  of  the  first  kind  (arithmetical  numbers)  are 
called  absolute  constants.  Constants  of  the  second  kind 
(letters)  are  called  arbitrary  or  general  constants. 

16.  By  general  agreement,  variables  are  represented  by 
the  last  letters  of  the  alphabet,  as  x,  y,  z;  while  constants 
are  represented  by  the  first  letters,  a,  b,  c  \  or  by  the  last 
letters  with  subscripts,  as  Xj,  y^,  x^,  y^,  etc. 

Exercise  5. 

1.  A  point  P  (x,  y)  revolves  about  the  point  Q  {x^,  yi), 
keeping  always  at  the  distance  a  from  it.  Name  the 
constants  and  the  variables  in  this  case.  What  is  the 
total  change  in  the  value  of  each  variable  ? 

2.  A  point  Q  {x,  y)  moves:  first  parallel  to  the  axis  of  //, 
then  parallel  to  the  axis  of  x,  then  equally  inclined  to  the 
axes.    Point  out  in  each  case  the  constants  and  the  variables. 


14 


ANALYTIC    GEOMETRY. 


Locus  OF  AN  Equation. 

17.  Let  us  continue  to  regard  x  and  y  as  the  coordinates 
of  a  point,  and  proceed  to  illustrate  the  meaning  of  an 
algebraic  equation  containing  one  or  both  of  these  letters. 

Take  as  the  first  case  the  equation  x  —  4=^0,  whence 
a;  =  4.  It  is  clear  that  this  equation  is  satisfied  by  the 
coordinates  of  every  point  so  situated  that  its  abscissa  is 
equal  to  4 ;  therefore,  it  is  satisfied  by  the  coordinate  of 

B 


Fig.  7. 

every  point  in  the  line  AB  (Fig.  7),  drawn  ||  to  OY,  on 
the  right  of  OY,  and  at  the  distance  4  from  OY.  And  it 
is  also  clear  that  this  line  contains  all  the  points  whose 
coordinates  will  satisfy  the  given  equation. 

The  line  AB,  then,  may  be  regarded  as  the  f/eometric 
representation  or  meaning  of  the  equation  x  —  4^0  ;  and, 
conversely,  the  equation  x  —  4  =  0  may  be  considered  to  be 
the  algebraic  representative  of  this  particular  line. 

In  Analytic  Geometry  the  line  AB  is  called  the  locus  of 
the  equation  x  —  4^0;  conversely,  the  equation  x  —  4  =  0 
is  known  as  the  equation  of  the  line  AB. 

The  line  AB  is  to  be  regarded  as  extending  indefinitely 
in  both  directions.  If  AB  is  described  by  a  point  P, 
moving  parallel  to  the  axis  of  y,  then  at  all  points  x  is 


LOCI    AND    THEIR    EQUATIONS. 


15 


constant  in  value  and  equal  to  4,  while  y  (which  does  not 
appear  in  the  given  equation)  is  a  variable,  passing  througli 
an  unlimited  number  of  values,  both  positive  and  negative. 

1 8.  The  equation  x  —  y  =  0,ov  x^y,  states  in  algebraic 
language  that  the  abscissa  of  the  point  is  always  equal  to 
the  ordinate. 


Values  of  x.                 Values  of  y. 
0 0. 

1 1. 

2     .....     2. 

—  1 -1. 

etc.  etc. 


Fig.  8. 

If  we  draw  through  the  origin  0  (Fig.  8)  a  straight 
line  AB,  bisecting  the  first  and  third  quadrants,  then  it  is 
easy  to  see  that  the  given  equation  is  satisfied  by  every 
point  in  this  line  and  by  no  other  points.  If  we  conceive  a 
point  P  to  move  so  that  its  abscissa  shall  always  be  equal 
to  its  ordinate,  then  the  point  must  describe  the  line  AB. 
In  other  words,  if  the  point  P  is  obliged  to  move  so  that 
its  coordinates  (which  of  course  are  variables)  shall  always 
satisfy  the  condition  expressed  by  the  equation  a-— //  =  0  ; 
then  the  motion  of  P  is  confined  to  the  line  AB. 

The  line  AB  is  the  locus  of  the  equation  x  —  y  =  0.  and 
this  equation  represents  the  line  AB. 

19.  The  equation  2x-\-y  —  3  =  0  is  satisfied  by  an  un- 
limited number  of  values  of  x  and  y.  We  may  find  as  many 
of  them  as  w.e  please  by  assuming  values  for  one  of  the  varia- 
bles, and  computing  the  corresponding  values  of  the  other. 


16 


ANALYTIC    GEOMETRY. 


If  we  assume  for  x  the  values  given  below,  we  easily  find 
for  y  the  corresponding  values  given  in  the  next  column. 

Values  of  x.  Values  of  y. 


0 

3. 

1 

1. 

2 

-1. 

3 

-3. 

4 

—  5. 

-1 

5. 

-2 

7. 

-3 

9. 

-4 

11. 

etc. 

etc. 

Fig.  9. 


Plotting  these  points  (as  shown  in  Fig.  9),  we  obtain  a 
series  of  points  so  placed  that  their  coordinates  all  satisfy 
the  given  equation.  By  assuming  for  x  values  between 
0  and  1,  1  and  2,  etc.,  we  might  in  the  same  way  obtain  as 
many  points  as  we  please  between  A  and  B,  B  and  C,  etc. 
In  this  case,  however,  the  points  all  lie  in  a  straight  line 
(as  will  be  shown  later);  so  that  if  any  tivo  points  are 
found,  the  straight  line  drawn  through  them  will  include 
all  the  points  whose  coordinates  satisfy  the  given  equa- 
tion. Now  imagine  that  a  point  P,  the  coordinates  of 
which  are  denoted  by  x  and  y,  is  required  to  move  in  such 
a  way  that  the  values  of  x  and  y  shall  always  satisfy  the 
equation  2x-\-y  —  3  =  0;  then  P  mtist  describe  the  line 
AB,  and  cannot  describe  any  other  line. 

The  line  AB  is  the  locus  of  the  equation  2x-^y  —  3  =  0. 

20.  Thus  far  we  have  taken  equations  of  the  first 
degree.  Let  us  now  consider  the  equation  x^  —  2/^  =  0. 
By  solving  for  y,  we  obtain  y  =  ±x.     Hence,   for  every 


LOCI    AND    THEIR    EQUATIONS. 


17 


value  of  X  there  are  tivo  values  of  y,  both  equal  numerically 
to  X,  but  having  unlike  signs.  Thus,  for  assumed  values 
of  X,  we  have  corresponding  values  of  y  given  below  : 


"Values  of  x. 

0  . 

1  . 
2 

3  . 

—  1  . 

—  2  . 

—  3  . 


Values  of  y. 

0. 

1,-1. 

2—2 

3,-3. 
-1,  1. 
-2,  2. 
-3,      3. 


Fig.  10. 


By  plotting  a  few  points,  and  comparing  this  case  with 
the  example  in  §  18,  it  becomes  evident  that  the  locus  of 
the  equation  consists  of  two  lines,  AB,  CD  (Fig.  10),  drawn 
through  the  origin  so  as  to  bisect  the  four  quadrants, 

21.  There  is  another  way  of  looking  at  this  case.  The 
equation  x'^  —  y^  =  0,  by  factoring,  may  be  written  (x  —  y) 
(x-]-y)^=  0.  Now  the  equation  is  satisfied  if  eithe?'  factor 
=  0;  hence,  it  is  satisfied  if  x  —  y  =  0,  and  also  if  x-\-y 
^0.  We  know  (see  §  18)  that  the  locus  of  the  equation 
x  —  y=^0  is  the  line  AB  (Fig.  8).  And  the  locus  of  the 
equation  x-\-y  =  0  (or  x  =  —  y)  is  evidently  the  line  CD, 
since  every  point  in  it  is  so  placed  that  the  two  coordinates 
are  equal  numerically  but  unlike  in  sign.  Therefore,  the 
original  equation  x'^  —  jf  =  0  is  represented  by  the  pair  of 
lines  AB  and  CD  (Fig.  10). 

22.  Let  us  next  consider  the  equation  cr^+?/  =  25. 
Solving  for  y,  we  obtain  y=±  V25  —  x\  When  x < 5 
there  are  two  values  of  y  equal  numerically  but  unlike  in 
sign.  When  x  =  5,  y  =  0.  When  a;  >  5  the  values  of  y 
are  imaginary ;  this  last  result  means  that  there  is  no  point 
with  an  abscissa  greater  than  5  whose  coordinates  will 
satisfy  the  given  equation. 


18 


ANALYTIC    GEOMETRY. 


By  assigning  values  of  x  differing  by  unity,  we  obtain 
tlie  following  sets  of  values  of  x  and  // ;  and  by  plotting 
the  points,  and  then  drawing  through  them  a  continuous 
curve,  we  obtain  the  curve  shown  in  Fig.  11. 


alues  of  X. 

Val 

aes  of  y. 

0      .      . 

±5. 

1      .      .      . 

±4.9. 

2     .     . 

±4.6. 

3     .     . 

±4. 

4     .     . 

±3. 

5     .     . 

0. 

-1     .     . 

±4.9. 

-2     .     . 

±4.6. 

—  3     .     . 

±4. 

—  4     .     . 

±3. 

-5     .     . 

0. 

In  this  case,  however,  the  locus  may  be  foiind  as  follows : 
Let  P  (Fig.  11)  be  any  point  so  placed  that  its  coordinates, 
x=OM,  i/  =  MP,  satisfy  the  equation  x^  -\-y-^^25.  Join 
OP;  then  x'-{-  f^'oF;  therefore,  OF  =  5.  Hence,  if  P  is 
any  point  in  the  circumference  described  with  0  as  centre 
and  5  for  radius,  its  coordinates  will  satisfy  the  given 
equation ;  and  if  P  is  not  in  this  circumference,  its  coor- 
dinates will  not  satisfy  the  equation.  This  circumference, 
then,  is  the  locus  of  the  equation. 

23.  The  points  whose  coordinates  satisfy  the  equation 
y^  =  4x  lie  neither  in  a  straight  line  nor  in  a  circumference. 
.Nevertheless,  they  do  all  lie  in  a  certain  line,  which  is, 
therefore,  completely  determined  by  the  equation.  To  con- 
struct this  line,  we  first  find  a  number  of  points  that  satisfy 
the  equation  (the  closer  the  points  to  one  another,  the 
better)  and  then  draw,  freehand  or  with  the  aid  of  tracing 
curves,  a  continuous  curve  through  the  points. 


LOCI    AND    THEIR    EQUATIONS. 


19 


The  coordinates  of  a  number  of  such  points  are  given  in 
the  table  below.  It  is  evident  that  for  each  positive  value 
of  X  there  are  two  values  of  ?/,  equal  numerically  but  un- 
like in  sign.  For  a  negative  value  of  x,  the  value  of  >/  is 
imaginary ;  this  means  that  there  are  no  points  to  the  left 
of  the  axis  of  y  that  will  satisfy  the  given  equation. 

Values  of  x.                     Values  of  y. 

•  0 0. 

1 ±2. 

2 ±2.83. 

3 ±3.46. 

4 ±4. 

5 ±4.47. 

6 ±4.90. 

7 ±5.29. 

8 ±5.66. 

9 ±6. 

—  1 imaginary. 

In  Fig.  12  the  several  points  obtained  are  plotted,  and  a 
smooth  curve  is  then  drawn  through  them.  It  passes 
through  the  origin,  is  placed  symmetrically  on  both  sides 
of  the  axis  of  x,  lies  wholly  on  the  right  of  the  axis  of  f/, 
and  extends  towards  the  right  without  limit.  It  is  the  locus 
of  the  given  equation,  and  is  a  curve  called  the  Parabola. 

24.  After  a  study  of  the  foregoing  examples,  we  may 
lay  down  the  following  general  principles,  which  form  the 
foundation  of  the  science  of  Analytic  Geometry: 

I.  Every  algebraic  equation  involving  x  and  i/  is  satis- 
fied by  an  unlimited  number  of  sets  of  values  of  x  and  i/; 
in  other  words,  x  and  //  may  be  treated  as  variaMes,  or 
quantities  varying  continuously,  yet  always  so  related  that 
their  values  constantly  satisfy  the  equation. 


Fig.  12. 


20  ANALYTIC    GEOMETRY. 

II.  The  letters  x  and  ?/  may  also  be  regarded  as  repre- 
senting the  coordinates  of  a  point.  This  point  is  not  fixed 
in  position,  because  x  and  y  are  variables ;  but  it  cannot  be 
placed  at  random,  because  x  and  y  can  have  only  such 
values  as  will  satisfy  the  equation;  now,  since  these  values 
are  continuous,  the  point  may  be  conceived  to  move  con- 
tinuously, and  will  therefore  describe  a  definite  line,  or 
group  of  lines. 

The  line,  or  group  of  lines,  described  by  a  point  moving 
so  that  its  coordinates  always  satisfy  the  equation  is  called 
the  Locus  of  the  Equation;  conversely,  the  equation  satis- 
fied by  the  coordinates  of  every  point  in  a  certain  line  is 
called  the  Equation  of  the  Line. 

An  equation,  therefore,  containing  the  variables  x  and  y 
is  the  algebraic  representation  of  a  line. 

In  Analytic  Geometry  the  loci  considered  are  represented 
by  their  equations,  and  the  investigation  of  their  properties 
is  carried  on  by  means  of  these  equations. 

Exercise  6. 

Determine  and  construct  the  loci  of  the  following  equa- 
tions (the  locus  in  each  case  being  either  a  straight  line  or 
a  circumference  of  a  circle) : 

1.  a;  — 6  =  0.  9.  9x-  — 25  =  0. 

2.  a;  +  5  =  0.  10.  ^x-—y'^  —  ^. 

3.  y=  —  l.  11.  a;2  — 16^2  =  0. 

4.  x  =  0.  12.  a;2_f_y2^36_ 

5.  ?/  =  0.  13.  x''-\-y''  —  l=Q. 

6.  x-\-y  =  0.  14.  x(?/H-5)=0. 

7.  x  —  2y  =  0.  15.  (x  — 2)(x  — 3)  =  G. 

8.  2x  +  3?/+10  =  0.  16.  (y-4)(3/+l)=0. 


LOCI    AND    TIIEIK    EQUATIONS.  21 

17.  What  is   the   geometric   meaning   of   the   equation 
5a;2  — 17x  — 12  =  0? 

Hint.     Resolve  the  equation  into  two  binomial  factors. 

18.  What  is   the   geometric  meaning   of   the  equation 

19.  What  two  lines    form  the  locus   of   the   equation 
xy-\-4:X^0? 

20.  Is  the  point  (2,  —  5)  situated  in  the  locus  of  the  equa- 
tion 4ic  —  3// -  22  =  0  ? 

Hint.     See  if  the  coordinates  of  the  point  satisfy  the  equation. 

21.  Is  the  point  (4,  —  6)  in  the  locus  of  the  equation 

22.  Is  the  point  ( —  1,  —  1)  in  the  locus  of  the  equation 
16a;2  +  V  +  15cc  —  6?/— 18  =  0  ? 

23.  Does  the  locus  of  the  equation  a---l-y"  =  100  pass 
through  the  point  ( — 6,  8)? 

24.  Which   of   the   loci  represented   by  the   following 
equations  pass  through  the  origin  ? 

(l)3x-[-2  =  0.  (5)3x  =  2i/. 

(2)3x-lUj-{-7  =  0.  {e)3x-n>/  =  0. 

(3)  a;2  -  16/  -10  =  0.  (7)  a-  —  1  Oy-  =  0. 

(4)  aa;  +  %  +  c  =  0.  (8)  ax-\-b>/  =  0. 

25.  The  abscissa  of  a  point  in  the  locus  of  the  equation 
Sx  —  4?/  —  7^0  is  9 ;  wliat  is  the  value  of  the  ordinate  ? 

Ans.  5. 

26.  Determine  that  point  in  the   locus  of   //  —  4a- ^0 

for  which  the  ordinate  =:  —  6. 

A]is.  The  point  (0,  -  6). 

27.  Determine  the  point  where  the  line  represented  by 
the  equation  lx-\-  y  — 14  =  0  cuts  tfie  axis  of  x. 

Ans.  The  point  (2,  0). 


22  analytic  geomexky. 

Intersections  of  Loci. 

25.  The  term  Curve,  as  used  in  Analytic  Geometry, 
means  any  geometric  locus,  including  the  straight  line  as 
well  as  lines  commonly  called  curves. 

The  Intercepts  of  a  curve  on  the  axes  are  the  distances 
from  the  origin  to  the  points  where  the  curve  cuts  the  axes. 

26.  To  find  the  intercepts  of  a  curve,  having  given  its 
equation. 

The  intercept  of  a  curve  on  the  axis  of  x  is  the  abscissa 
of  the  point  where  the  curve  cuts  the  axis  of  x.  The 
ordinate  of  this  point  =  0.  Therefore,  to  find  this  inter- 
cept, put  v/  =  0  in  the  given  equation  of  the  curve,  and 
then  solve  the  equation  for  x ;  the  resulting  real  values  of 
X  will  be  the  intercepts  required. 

If  the  equation  is  of  a  higher  degree  than  the  first,  there 
will  in  general  be  more  than  one  real  value  of  x;  and  the 
curve  will  intersect  the  axis  of  x  in  as  many  points  as  there 
are  real  values  of  x. 

To  an  imaginary  value  of  x  there  corresponds  no  inter- 
cept ;  but  it  is  sometimes  convenient  to  speak  of  such  a 
value  as  an  imaginary  intercept. 

Similarly,  to  find  the  intercepts  on  the  axis  of  y,  put 
a;  =  0  in  the  given  equation,  and  then  solve  it  for  y ;  the 
resulting  real  values  of  y  will  be  the  intercepts  required. 

27.  To  find  the  points  of  intersection  of  two  curves,  having 
given  their  equations. 

Since  the  points  of  intersection  lie  in  both  curves,  their 
coordinates  must  satisfy  both  equations.  Tlierefore,  to 
find  their  coordinates,  solve  the  two  equations,  regarding 
the  variables  x  and  y  as  unknown  quantities. 

If  the  equations  are  both  of  the  first  degree,  there  will 


LOCI    AND    THEIR    EQUATIONS.  23 

be  only  one  pair  of  values  of  x  and  y,  and  one  point  of 
intersection. 

If  the  equations  are,  one  or  both  of  them,  of  higher 
degree  than  the  first,  there  may  be  several  pairs  of  values 
of  X  and  y ;  in  this  case  there  will  be  as  many  points  of 
intersection  as  there  are  pairs  of  real  values  of  x  and  y. 

If  imaginary  values  of  either  x  or  y  are  obtained,  there 
are  no  corresponding  points  of  intersection. 

28.  If  a  curve  2>asses  throxujh  the  oriyin,  its  equation,  re- 
duced to  its  slmjdest  form,  cannot  have  a  constant  term;  that 
is,  cannot  have  a  term  free  from  both  x  and  y. 

Since  in  this  case  the  point  (0,  0)  is  a  point  of  the  curve, 
its  equation  must  be  satisfied  by  the  values  cc  =  0,  and 
y^=0.  But  it  is  obvious  that  these  values  cannot  satisfy 
the  equation  if,  after  reduction  to  its  simplest  form,  it  still 
contains  a  constant  term.  Therefore  the  equation  cannot 
have  a  constant  term. 

29.  If  an  equation  has  no  constant  term,  its  locus  must 
pass  through  the  origm. 

For,  the  values  x  =  0,  y  =  0  must  evidently  satisfy  the 

equation,  and  therefore  the  point  (0,  0)  must  be  a  point  of 

the  locus. 

Exercise  7. 

Find  the  intercepts  of  the  following  curves  : 

1.  4a-  +  3//  — 48  =  0.  8.  0-  — 3  =  0. 

2.  5y  — ox  — 30^0.  9.  a-  — 9=0. 

3.  ic2  +  2/2  =  16.  10.  a-'-  — //-  =  0. 

4.  9a- +  4/ =  16.  11.  /  =  4a-. 

5.  9.r2  — 4/  =  16.  12.  .T-  +  //  — 4.r  — 8//  =  32. 

6.  9x^  —  4//  =  1 6.  13.  X-  +  //  —  4.r  —  8//  =  0. 

7.  a:'x' -\- by  =  a'b\  14.  (x  — 5)' -\- (y —  ())'  =  20. 


24  ANALYTIC    GEOMETKY. 

Find  the  points  of  intersection  of  the  following  curver; 

15.  Sx  — 4:1/ -{-13  =  0,  lla;+7^— 104=0. 

16.  2a; +  3^  =  7,  x—i/  =  l. 

17.  a;-7//  +  25  =  0,  x'-\-i/^  =  25. 

18.  3x-j-ii/  =  2o,  x'-\-i/  =  25. 

19.  x-\-y  =  8,  x--{-f  =  34.. 

20.  2x=i/,  x^  +  i/^  —  10x  =  0. 

21.  The  equations  of  the  sides  of  a  triangle  are  2x-{- 
9y  +  17=r0,  7a;  — ^—38  =  0,  a;  — 2^-^2  =  0.  Find  the 
coordinates  of  its  three  vertices. 

22.  The  equations  of  tlie  sides  of  a  triangle  are  5a;  +  6^ 
=  12,  3a;  —  4^  =  30,  a;  +  5y  =  10.  Find  the  lengths  of  its 
sides. 

23.  Find  the  lengths  of  the  sides  of  a  triangle  if  the 
equations  of  the  sides  are  a;  =  0,  //^O,  and  4a; +  3^  =  12. 

24.  What  are  the  vertices  of  the  quadrilateral  enclosed 
by  the  straight  lines  a;  —  a  =  0,  a;  +  a  =  0,  y  —  b  =  0,  y-\-b 
=  0?     What  kind  of  a  quadrilateral  is  it? 

25.  Does  the  straight  line  5a;  +  4y  =  20  cut  the  circle 

26.  Find  the  length  of  that  part  of  the  straight  line 
3a;  —  4y  =  0  which  is  contained  within  the  circle  x^-\-if 
=  25. 

27.  Which  of  the  following  curves  pass  through  the 
origin  of  coordinates? 

(1)  7a;  — 2y+4  =  0.  (4)  ax-\-by  =  0. 

(2)  7a;  — 2//  =  0.  (5)  ax  +  %  +  c  =  0. 

(3)  tf  —  x'  =  4:y.  (6)  x'^  —  y-\-a  =  a-\-xy. 

28.  Change  the  equation  4a'H-2//  — 7^0  so  that  its 
locus  shall  pass  through  the  origin. 


LOCI    AND    THEIK    EQUATIONS,  25 

CONSTKUCTION    OF    LoCI. 

30.  If  we  know  that  the  locus  of  a  given  equation  is  a 
straight  line,  the  locus  is  easily  constructed;  it  is  only 
necessary  to  find  any  two  points  in  it,  plot  them,  and  draw 
a  straight  line  through  them  with  the  aid  of  a  ruler. 

Likewise,  if  we  know  that  the  locus  is  a  circumference, 
and  can  find  its  centre  and  its  radius,  the  entire  locus  can 
then  be  described  with  the  aid  of  a  pair  of  compasses. 

It  will  appear  later  that  the  form  of  the  given  equation 
enables  us  at  once  to  tell  whether  its  locus  is  a  straight  line 
or  a  circumference. 

If  the  locus  of  an  equation  is  neither  a  straight  line  nor 
a  circumference,  then  the  following  method  of  construction, 
which  is  applicable  to  the  locus  of  any  equation  without 
regard  to  the  form  of  the  curve,  is  usually  employed. 

31.  To  construct  the  locus  of  a  given  equation. 
The  steps  of  the  process  are  as  follows: 

1.  Solve  the  equation  with  respect  to  either  x  or  y. 

2.  Assign  values  to  the  other  variable,  diifering  not 
mucli  from  one  another. 

3.  Find  each  corresponding  value  of  the  first  variable. 

4.  Draw  two  axes,  choose  a  suitable  scale  of  lengths,  and 
plot  the  points  whose  coordinates  have  been  obtained. 

5.  Draw  a  continuous  curve  through  these  points. 
Discussion.     An  examination  of  the  equation,  as  shown 

in  the  examples  given  below,  enables  us  to  obtain  a  good 
general  idea  of  the  shape  and  size  of  the  curve,  its  position 
with  respect  to  the  axes,  etc. ;  in  this  way  it  serves  as  an  aid 
in  constructing  the  curve,  and  as  a  means  of  detecting  nu- 
merical errors  made  in  computing  the  coordinates  of  the 
points.  Such  an  examination  is  called  a  discussion  of  the 
equation. 


26 


ANALYTIC    GEOMETRY. 


Note  1.  This  method  of  constructing  a  locus  is  from  its  nature  an 
approximate  metliod.  But  the  nearer  tlie  points  are  to  one  another, 
tlie  nearer  the  curve  will  approacli  the  exact  position  of  the  locus. 

Note  2.  In  theory,  it  is  immaterial  what  scale  of  lengths  is  used. 
In  practice,  the  unit  of  lengths  should  be  determined  by  tlie  size  of  the 
paper  compared  with  the  greatest  length  to  be  laid  off  upon  it.  Paper 
sold  under  the  name  of  "  coordinate  paper,"  ruled  in  small  squares,  j^^ 
of  an  inch  on  a  side,  will  be  found  very  convenient  in  practice. 

32.    Construct  the  locus  of  the  equation 

9^2 +  4/ -576  =  0. 

If  we  solve  for  both  x  and  y,  we  obtain  the  following  values : 


'=:±aV64-; 


;  =  ±§Vl44  — ?/2 


(1) 
(2) 


By  assigning  to  x  values  differing  by  unity,  and  finding 
corresponding  values  of  ?/,  we  obtain  the  results  given  below. 
To  each  value  of  x,  positive  or  negative,  there  correspond  two 
values  of  y,  equal  numerically  and  unlike  in  sign.  By  plotting 
the  corresponding  points,  and  drawing  a  continuous  curve 
through  them,  we  obtain  the  closed  curve  shown  in  Fig.  13. 


Values  of  x 

Values  of  y. 

0 

.     .     ±  12. 

±1 

±  11.91. 

±2 

±11.62. 

±3 

±11.13. 

±4 

±  10.39. 

±5 

±   9.36. 

±6 

±   7.93. 

±7 

±    5.80. 

±8 

±    0. 

±9 

±  imaginary 

LOCI    AND    THEIR    EQUATIONS.  27 

Discussion.  From  equations  (1)  and  (2)  we  see  that 
if  X  =0,  ij  =  ±i  12,  and  if  y  =  0,  x  =  ±:8;  therefore,  the 
intercepts  of  the  curve  on  the  axis  of  x  are  +8  and  — 8, 
and  those  on  the  axis  of  y  are  + 12  and  — 12.  These  inter- 
cepts are  the  lengths  OA,  OA',  and  OB,  OB',  in  Eig.  13. 

If  we  assign  to  a?  a  numerical  value  greater  than  8,  posi- 
tive or  negative,  we  find  by  substitution  in  equation  (1)  that 
the  corresponding  value  of  ij  will  be  imaginary.  This  shows 
that  OA  and  OA'  are  the  maximum  abscissas  of  the  curve. 
Similarly,  equation  (2)  shows  that  the  curve  has  no  points 
with  ordinates  greater  than  +12  and  — 12. 

The  greater  the  numerical  value  of  x,  between  the  limits 
0  and  4"  8  or  0  and  —  8,  the  less  the  corresponding  value  of 
y  numerically  ;    why  ? 

From  equation  (1)  we  see  that  for  each  value  of  x, 
between  the  limits  0  and  i  8,  there  are  two  real  values  of 
y,  equal  numerically  and  unlike  in  sign.  Hence,  for  each 
value  of  X  between  0  and  ±  8  there  are  two  points  of  the 
curve  placed  equally  distant  from  the  axis  of  x.  Therefore, 
the  curve  is  symmetrical  with  respect  to  the  axis  of  x  ;  iu 
other  words,  if  the  portion  of  the  curve  above  the  axis  of  x 
is  revolved  about  this  axis  through  180°,  it  will  coincide  with 
the  portion  below  the  axis.  Similarly,  it  follows  from  equa- 
tion (2)  that  the  curve  is  also  symmetrical  with  respect  to 
the  axis  of  y.  Therefore,  the  entire  curve  is  a  closed  curve, 
consisting  of  four  equal  quadrantal  arcs  symmetrically  placed 
about  the  origin  0.     The  name  of  this  curve  is  the  Ellipse. 

33.    Construct  the  locus  of  the  equation 

4x  — /+16  =  0. 
Solving  for  both  x  and  y/,  we  obtain 

y=±2^ir^,  (1) 

-^-^-  (2) 


28 


ANALYTIC    GEOMETRY. 


We  may  either  assign  values  to  x,  and  then  compute 
values  of  y  by  means  of  (1),  or  assign  values  to  y,  and  com- 
pute values  of  x  by  means  of  (2)  ;  the  second  course  is 
better,  because  there  is  less  labor  in  squaring  a  number 
than  in  extracting  its  square  root. 

By  assigning  values  to  y,  differing  by  unity  from  0  to 
+ 10,  and  from  0  to  — 10,  and  then  proceeding  exactly  as 
in  the  last  example,  we  obtain  the  series  of  values  given 
below,  and  the  curve  shown  in  J'ig  14. 

Values  of  y. 
±0 


Fig.  14. 


±1 

±2 
±3 

±4 

±5 
±6 


t9 
10 


Values  of  x. 

■   -4. 

—  3.7o. 

-3. 

-1.75. 

0. 

2.25. 
5. 

8.25. 
12. 

16.25. 
21. 

Discussiox.  An  examination  of  equations  (1)  and  (2) 
yields  the  following  results,  the  reasons  for  which  are  left 
as  an  exercise  for  the  learner : 

The  intercepts  on  the  axes  are  : 
On  the  axis  of  x,         OA  =  —  4. 
On  the  axis  of  y,         0B  =  +  4,    and  6»C=  —  4. 

If  we  draw  through  A  the  line  AD  _\_  to  OX,  the  entire 
curve  lies  to  the  right  of  AD. 

The  curve  is  situated  on  both  sides  of  OX,  and  is  sym- 
metrical with  res})ect  to  OX. 

The  curve  extends  towards  the  right  without  limit. 


LOCI    AND    THEIR    EQUATIONS. 


29 


The  curve  constantly  recedes  from  OX  as  it  extends 
towards  the  right. 

This  curve  is  called  a  Parabola  ;  the  point  A  is  called  its 
Vertex  ;   the  line  AX  its  Axis. 

34.    Construct  the  locus  of  the  equation 
y  =  sin  X. 

If  we  assume  for  x  the  values  0°,  10°,  20°,  30°,  etc.,  the 
corresponding  values  of  >/  are  the  natural  sines  of  these 
angles,  and  are  as  follows  : 


Values  of 

X.                Values  of  y. 

Values  of  x. 

Values  of  y 

0° 

...     0. 

50°     .     . 

.     .     0.77. 

10° 

.     .     .     0.17. 

60°     .     . 

.     .     0.87 

20° 

.     .     .     0.34. 

70°     .     . 

.     .     0.94. 

30° 

.     .     .     0.50. 

80°     .     . 

.     .     0.98. 

40° 

.     .     .     0.64. 

90°     .     . 

.     .     1. 

If  we  continue  the  values  of  x  from  90°  to  180°,  the 
above  values  of  y  repeat  themselves  in  the  inverse  order 
(e.g.,  if  a;  =  100°,  y  =  0.98,  etc.)  ;  from  180°  to  360°  the 
values  of  y  are  numerically  the  same,  and  occur  in  the 
same  order  as  between  0°  and  180°,  but  are  negative. 


Fig.  15. 


In  order  to  express  both  x  and  y  in  terms  of  a  common 
linear  unit,  we  ought,  in  strictness,  to  use  the  circular  meas- 
ure of  an  angle  in  which  the  linear  unit  represents  an  angle 


30  ANALYTIC    GEOMETRY. 

of  57.3°,  very  nearly  (see  §  5).  But  it  is  more  convenient, 
and  serves  our  present  purpose  equally  well,  to  assume  that 
an  angle  of  G0°  =  the  linear  unit.  This  assumption  is  made 
in  Fig.  15,  where  the  curve  is  drawn  with  one  centimeter 
as  the  linear  unit. 

Discussion.  The  curve  passes  through  the  origin,  and 
cuts  the  axis  of  x  at  points  separated  by  intervals  of 
180°.  Since  an  angle  may  have  any  magnitude,  positive  or 
negative,  the  curve  extends  on  both  sides  of  the  origin 
without  limit.  •  The  maximum  value  of  the  ordinate  is 
alternately  + 1  and  —  1  :  the  former  value  corresponds  to 
the  angle  90°,  and  repeats  itself  at  intervals  of  360°;  the 
latter  value  corresponds  to  the  angle  270°,  and  repeats 
itself  at  intervals  of  360°.  The  curve  has  the  form  of  a 
wave,  and  is  called  the  Sinusoid. 

Exercise  8. 

Construct  the  loci  of  the  following  equations  : 


1. 

3x  —  y  —  2  ==  0. 

13. 

r-i=o. 

2. 

y  =  2x. 

14. 

y  =  x\ 

3. 

x'^^y\ 

15. 

xy  =  12. 

4. 

^2+,/ =100. 

16. 

X  ■=  sin  y. 

5. 

^2-^=^  =  25. 

17. 

y  =^  2  sin  x. 

6. 

4a;2-/r=0. 

18. 

y  =^  sin  2x. 

7. 

4a;2+V  =  144. 

19. 

2/  =  cos  X. 

8. 

y  — 16x  =  0. 

20. 

y  =  tan  x. 

9. 

y/2_|_  1(5^=0. 

21. 

y  =  cot  x. 

10. 

.T^— 2.T  — lOy  — 5==0. 

22. 

y  =  sec  x. 

11. 

y--2y-10a-  =  0. 

23. 

y  =^  CSC  X. 

12. 

(.«-3)^+(/y-2y^=:25. 

24. 

y  =  sin  X  +  cos  x. 

loci  and  theik  equatioxs.  31 

Equation  of  a  Curve. 

35.  From  what  precedes,  we  may  conclude  that  every 
equation  involving  x  and  y  as  variables  represents  a 
definite  line  (or  group  of  lines)  known  as  the  locus  of  the 
equation.  Regarded  from  this  point  of  view,  an  equation 
is  the  statement  in  algebraic  language  of  a  geometric  con- 
dition which  must  always  be  satisfied  by  a  point  (x,  y),  as 
we  imagine  it  to  move  in  the  plane  of  the  axes.  For  ex- 
ample, the  equation  x^=2y  states  the  condition  that  the 
point  must  so  move  that  its  abscissa  shall  always  be  equal 
to  twice  its  ordinate;  the  equation  x^ -[-1/^=4:  states  the 
condition  that  the  point  must  so  move  that  the  sum  of  the 
squares  of  its  coordinates  shall  always  be  equal  to  4 ;  etc. 

Conversely,  every  geometric  condition  that  a  point  is 
required  to  satisfy  must  confine  the  point  to  a  definite  line 
as  its  locus,  and  must  lead  to  an  equation  that  is  always 
satisfied  by  the  coordinates  of  the  point. 

Hence  arises  a  new  problem,  and  one  usually  of  greater 
difficulty  than  any  thus  far  considered,  namely: 

Given  the  geometric  condition  to  be  satisfied  by  a  point,  to 
find  the  equation  of  its  locus. 

The  importance  of  this  problem  is  that  in  the  practical 
applications  of  Analytic  Geometry  the  law  of  a  moving 
point  is  commonly  the  one  thing  known,  so  that  the  first 
step  must  consist  in  finding  the  equation  of  its  locus. 

Exercise  9. 

1.  A  point  moves  so  that  it  is  always  three  times  as  far 
from  the  axis  of  x  as  from  the  axis  of  y.  What  is  the 
equation  of  its  locus? 

2.  "What  is  the  equation  of  the  locus  of  a  point  that 
moves  so  that  its  abscissa  is  always  equal  to +  6?    —  6?    0? 


32  ANALYTIC    GEOMETRY. 

3.  What  is  the  e(;[uation  of  the  locus  of  a  point  that 
moves  so  that  its  ordinate  is  always  equal  to -f- 4  ?    — 1?    0? 

4.  A  point  so  moves  that  its  distance  from  the  straight 
line  X  ==  3  is  always  numerically  equal  to  2.  What  is  the 
equation  of  its  locus  ? 

5.  A  point  so  moves  that  its  distance  from  the  straight 
line  y  =  5  is  always  numerically  equal  to  3.  Find  the 
equation  of  its  locus.     Construct  the  locus. 

6.  A  point  moves  so  that  its  distance  from  the  straight 
lina  X  +  4  ^  0  is  always  numerically  equal  to  5.  Find  the 
equation  of  its  locus.     Construct  the  locus. 

7.  What  is  the  equation  of  the  locus  of  a  point  equidistant 

(1)  from  the  parallels  x  =  0  and  x  ==  —  6  ? 

(2)  from  the  parallels  y=^l  and  ?/  =  —  3? 

8.  What  is  the  equation  of  the  locus  of  a  point  always 
equidistant  from  the  origin  and  the  point  (6,  0)  ? 

Find  the  equation  of  the  locus  of  a  point 

9.  Equidistant  from  the  points  (4,  0)  and  (—  2,  0). 

10.  Equidistant  from  the  points  (0,  —  5)  and  (0,  9). 

11.  Equidistant  from  the  points  (3,  4)  and  (5,  —  2). 

12.  Equidistant  from  the  points  (5,  0)  and  (0,  5). 

13.  A  point  moves  so  that  its  distance  from  the  origin 
is  always  equal  to  10.     Find  the  equation  of  its  locus. 

14.  A  point  moves  so  that  its  distance  from  the  point 
(4,-3)  is  always  equal  to  5.  Find  the  equation  of  its 
locus,  and  construct  it.  What  kind  of  curve  is  it?  Does 
it  pass  through  the  origin?     Why  ? 

15.  What  is  the  equation  of  the  locus  of  a  point  whose 
distance  from  the  point  (—  4,  —  7)  is  always  equal  to  8  ? 


LOCI    AND    THEIR    EQUATIONS.  33 

16.  About  the  origin  of  coordinates  as  centre,  with  aradius 
equal  to  5,  a  circle  is  described.  A  point  outside  this  circle 
so  moves  that  its  distance  from  the  circumference  of  the  cir- 
cle is  always  equal  to  4.     What  is  the  equation  of  its  locus? 

17.  A  high  rock  A,  rising  out  of  the  water,  is  3  miles 
from  a  perfectly  straight  shore  BC.  A  vessel  so  moves  that 
its  distance  from  the  rock  is  always  the  same  as  its  distance 
from  the  shore.     What  is  the  equation  of  its  locus? 

18.  A  point  A  is  situated  at  the  distance  6  from  the 
line  BC.  A  moving  point  P  is  always  equidistant  from 
A  and  BC.     Find  the  equation  of  its  locus. 

19.  A  point  moves  so  that  its  distance  from  the  axis  of  x  is 
half  its  distance  from  the  origin ;  find  the  equation  of  its  locus. 

20.  A  point  moves  so  that  the  sum  of  the  squares  of  its 
distances  from  the  two  fixed  points  (a,  0)  and  ( — a,  0)  is 
the  constant  2k^;  find  the  equation  of  its  locus. 

21.  A  point  moves  so  that  the  difference  of  the  squares 
of  its  distances  from  (a,  0)  and  ( — a,  0)  is  the  constant  k^; 
find  the  equation  of  its  locus. 

Exercise  10.     (Review^.) 

1.  If  we  plot  all  possible  points  for  which  x  =  — 5, 
how  will  they  be  situated  ? 

2.  Construct  the  point  (x,  y)  if  a-  =  2  and 

(l)7/  =  4x-3,     (2)3x-2y-8. 

3.  The  vertices  of  a  rectangle  are  the  points  (a,  b),  ( —  a, 
b),  ( — a,—b),  and  (a, — b).  Find  the  lengths  of  its  sides, 
the  lengths  of  its  diagonals,  and  show  tliat  the  vertices 
are  equidistant  from  the  origin. 

4.  What  does  equation  [1].  p.  6,  for  the  distance  between 
two  points,  become  when  one  of  the  points  is  the  origin  ? 


34  ANALVTIO    GEOMETRY. 

5.  Express  by  an  equation  that  the  distance  of  the  point 
(x,  if)  from  the  point  (4,  6)  is  equal  to  8. 

6.  Express  by  an  equation  tliat  the  point  (x,  y)  is  equi- 
distant from  the  i)oints  (2,  3)  and  (4,  5). 

7.  Find  the  point  equidistant   from  the  points  (2,  3), 
(4,  5),  and  (6,  1).     What  is  the  common  distance? 

8.  Prove  that  the  diagonals  of  a  rectangle  are  equal. 

9.  Prove  that  the  diagonals  of  a  parallelogram  mutually 
bisect  each  other. 

10.  The  coordinates  of  three  vertices  of  a  parallelogram 
are  known :  (5,  3),  (7,  10),  (13,  9).  What  are  the  coordi- 
nates of  the  remaining  vertex  ? 

11.  The  coordinates  of  the  vertices  of  a  triangle  are 
(3,5),  (7,-9),  (2,-4).  Find  the  coordinates  of  the 
middle  points  of  its  sides. 

12.  The  centre  of  gravity  of  a  triangle  is  situated  on  the 
line  joining  any  vertex  to  the  middle  point  of  the  opposite 
side,  at  the  point  of  triseetion  nearer  tliat  side.  Find  the 
centre  of  gravity  of  the  triangle  whose  vertices  are  the 
points  (2,  3),  (4,-5),  (-3,-6). 

13.  Tlie  vertices  of  a  triangle  are  (5,  —  3),  (7,  9),  (—9,  6). 
Find  the  distance  from  its  centre  of  gravity  to  tlie  origin. 

14.  If  tlie  vertices  of  a  quadrilateral  are  (0,  0),  (5,  0), 
(9,  11),  (0,3),  what  are  the  coordinates  of  the  intersection 
of  the  two  straight  lines  that  join  the  middle  points  of  the 
opposite  sides  ? 

15.  Prove  that  the  two  straight  lines  which  join  the 
middle  points  of  tlie  opposite  sides  of  any  quadrilateral 
mutually  bisect  each  other. 

16.  A  line  is  divided  into  three  equal  parts.  One  end  of 
the  line  is  the  point  (3,  8)  ;  the  adjacent  point  of  division 
is  (4,  13).     What  are  the  coordinates  of  the  other  end? 


LOCI    AND    THEIR    EQUATIONS.  35 

17.  The  line  juiiiiiig  the  jnjiuts  (xi.  y^)  and  (x^,  y^)  is 
divided  into  four  equal  parts.  Find  the  coordinates  of 
the  points  of  division. 

18.  Explain  and  illustrate  the  relation  tliat  exists 
between  an  equation  and  its  locus. 

19.  Construct  the  two  lines  that  form  the  locus  of  the 
equation  x"  —  7a"  =;  0. 

20.  Is  the  point  (2,  ~5)  in  the  locus  of  the  equation 
4a:=^  — V  =  36? 

21.  Tlie  ordinate  of  a  certain  point  in  the  locus  of  the 
equation  .r-  +  if  -\-  20x'  —  70  =;  0  is  1.  What  is  the  abscissa 
of  this  point? 

22.  Find  the  intercepts  of  the  curve 

a:^  +  y— 5..-7y  +  6  =  0. 
Find  the  points  common  to  the  curves : 

23.  x'-\-y^  =  100,  and  y-  —  —  =  ^- 

24.  a;^  +  //^  =^  5«^,  and  ic^  =  4«7/. 

25.  U^x"  +  ay  =  tt^W,  and  a;^  +  /  =  a\ 

26.  Find  the  lengths  of  the  sides  of  a  triangle,  if  its 
vertices  are  (6,  0),  (0,  —8),  (—4,  —2). 

27.  A  point  moves  so  that  it  is  always  six  times  as  far 
from  one  of  two  fixed  perpendicular  lines  as  from  the 
other.     Find  the  equation  of  its  locus. 

28.  A  point  so  moves  that  its  distance  from  the  fixed 
point  A  is  always  double  its  distance  from  the  fixed  line 
AB.     Find  the  equation  of  its  locus. 

29.  A  fixed  point  is  at  the  distance  a  from  a  fixed 
straight  line.  A  point  so  moves  that  its  distance  from  the 
fixed  point  is  always  twice  its  distance  from  the  fixed  Hue. 
Find  the  equation  of  its  locus. 


CHAPTER  II. 
THE   STRAIGHT  LINE. 

Equations  of  the  Straight  Line. 

36.  Notation.  Throughout  this  chapter,  and  generally 
in  equations  of  straight  lines, 

a  =  the  intercept  on  the  axis  of  x. 
h  =  the  intercept  on  the  axis  of  y. 
y  =  the  angle  between  the  axis  of  x  and  the  line, 
m  =  tan  y. 

p  =  the  perpendicular  from  the  origin  to  the  line. 
a  =  the  angle  between  the  axis  of  x  and  p. 

These  six  quantities  are  general  constants;  a,  b,  and  7n 
may  have  any  values  from  —  ao  to  +  cc  ;  ^,  any  value 
from  0  to  -)-  oc  ;  y,  any  value  from  0°  to  180°  ;  a,  any 
value  from  0°  to  360°. 

The  constant  m  is  often  called  the  Slope  of  the  line ; 
its  value  determines  the  direction  of  the  line. 

In  order  to  determine  a  straight  line,  two  geometric 
conditions  must  be  given. 

37.  To  find  the  equation  of  a  straight  line  passing  through 
two  given  points  (x-^,  y^  and  (x2,  y^. 

Let  A  (Fig.  16)  be  the  point  (xj,  y-^,  B  the  point 
(^2,  2/2)  ;  and  let  P  be  any  point  of  tlie  line  drawn  through 
A  and  B,  x  and  ?/  its  coordinates.  Draw  AC,  BD,  PM  \\ 
to  OY,  and  AEF  \\  to  OX. 


THE    STRAIGHT    LIXE. 


37 


The  triangles  APF,  ABE  are  similar;  therefore, 
PF  _BE 
AF~  AE 


A^iow,  FF—  1/  —  j/i,  AF=x  —  Xi,  BE=i/o—  >/i,  AE=X2  —  Xi. 

Therefore,  ^^-^^^  ^^^^^y  W 

This  is  the  equation  required. 

Y 


Fig.  16. 


Fig.  17. 


It  is  evident  that  the  angle  FAP  =  y.  Therefore,  each 
side  of  equation  [4]  is  equal  to  tan  y  or  m.  The  first  side 
contains  the  two  variables  x  and  ?/,  and  the  equation  tells  us 

that  they  must  vary  in  such  a  way  that  the  fraction  ' — 

remains  constant  in  value  and  equals  m. 

Note.  In  Fig.  16  the  points  A,  B,  and  P  are  assumed  in  the  first 
quadrant  in  order  to  avoid  negative  quantities.  But  the  reasoning  will 
lead  to  equation  [4]  whatever  be  the  positions  of  ,these  points.  In 
Fig.  17  the  points  are  in  different  quadrants.  The  triangles  APF, 
ABE  are  to  be  constructed  as  shown  in  the  figure.  They  are  similar  ; 
and  by  taking  proper  account  of  the  algebraic  signs  of  the  quantities, 
we  arrive  at  equation  [4],  as  before.  The  learner  should  study  this 
case  with  care,  and  should  study  other  cases  devised  by  himself,  till 
he  is  convinced  that  equation  [4]  is  perfectly  general. 


38 


ANALYTIC    (JEOMETRV. 


38.  To  find  the  equation  of  a  straujld  line,  given  one 
point  (xy,  ?/i)  in  the  line  and  the  slope  ni. 

Let  the  figure  be  constructed  like  Fig.  16,  omitting  the 
point  B  and  the  line  BED.     Then  it  is  evident  that 


whence, 


PF 


y—yx 


AF     ic  — aji' 
y  —  y\  =  tn(3c  —  oci). 


Y 

ir 

I 

Y 

c 

y      () 

J 

I     X 

Y 


[5] 


Fig.  18. 


M  A\  X 


Fig.  19. 


39.  To  find  the  equation  of  a  strai/jht  line,  gicen  the 
intercept  h  and,  the  angle  y. 

Let  the  line  cut  the  axes  in  the  points  A  and  B  (Fig.  18). 
Let  P  be  any  point  (x,  y)  in  tlie  line.  Draw  PM^  to  OY, 
and7?C||  to  OX. 

Then  OB  =  h,  rBC  =  y,  BC  =  x,  PC  =  y  —  b; 

therefore,  m  = ' : 

X 

whence,  y  =  utx  +  h.  [6] 

40.  To  find  the  equation  of  a  straight  line,  given  its  inter- 
cepts  a  and  h. 

Let  the  line  cut  the  axes  in  tlie  points  A  and  B  (Fig.  19), 
and  let  P  be  any  point  (x,  y)  in  the  line.     Then  OA  =  a, 


THE    STKAIGHT    LINK. 


39 


OB  =  h.     Drawi^J/Xtu  OX.     The  triangles  F MA,  BOA 
are  similar  ;  therefore, 

PM      MA       OA  —  OM 


J 

OA              OA 

y  - 
b 

a  —  X      ^       X 

J-          5 
a                  a 

a      b 

whence,  7;  +  a  ~  -'^  [^] 

This  is  called  the  Symmetrical  Equation  of  the  straight 
line. 

41.  To  find  the  e(ivat\oii  of  a  straight  line,  given  its  dis- 
tance p  from  the  origin,  and  tlce  angle  a. 


i 

/>' 

vS 

r/ 

/ 

'*' 

\r 

\^i 

0 

.1/ 

\.Y 

Fig.  20. 

Let  AB  (Fig.  20)  be  the  line,  P  any  jioint  in  it.  Draw 
OS  L  to  AB,  meeting  AB  in  ,S;  7^J/ _L  to  OA';  Mii  ||  to 
AB,  meeting  OS  in  U\  ami  /'^  J_  to  All. 

Tlien  p=OS=  Oil  +  QB,  a  =  XOS  =  PMQ. 

Now,  OR  —  OM  cos  a  =  .^'  cos  a, 

and  QP  =  PM  sin  a  —  y  sin  a. 

Therefore,        OR  +  (>F  =;>  =  .r  cos  a  +  v/  sin  a, 
or  -z^  cos  a  +  //  sin  a  =  7>.  [8] 

This  is  called  the  Normal  Equation  of  the  straight  line. 

The  coefficients  cos  a  and  sin  a  determine  the  dii-ection 
of  the  line,  and  p  its  distance  from  the  origin. 


40  AXALYTIC    GEOMETRY. 

Note.  Observe  that  all  the  equations  of  the  straight  line  that 
have  been  obtained  are  of  the  first  degree.  Their  differences  in  form 
are  due  to  the  constants  which  enter  them.  The  form  of  each,  and 
the  signification  of  its  constants,  should  be  thoroughly  fixed  in  mind. 

Exercise  11. 

Find  the  equation  of  the  straight  line  passing  through 
the  two  points  : 

1.  (2,  3)  and  (4,  5).  7.  (2,  5)  and  (0,  7). 

2.  (4,  5)  and  (7,  11).  8.  (3,  4)  and  (0,  0). 

3.  (—  1,  2)  and  (3,  —  2).  9.  (3,  0)  and  (0,  0). 

4.  (—2,-2)  and  (-3, -3).  10.  (3,  4)  and  (—2,  4). 

5.  (4,  0)  and  (2,  3).  11.  (2,  5)  and  (-2,-5). 

6.  (0,  2)  and  (—  3,  0).  12.   (m,  7i)  and  (—  m,  —  n). 

Find  the  equation  of  a  straight  line,  given  : 

13.  (4,  1)  andy  =  45^  2*9.  b  =  —4,y=120°. 

14.  (2,  7)  and  y  =  60°.  30.  6  = -4,  y  =  135°. 

15.  (— 3,  11)  andy  =  45°.  31.  b  =  —  4=,  y  =  150°. 

16.  (13,-4)  and  y=:.150°.  32.  b  =  —  4:,  y  =  180°. 

17.  (3,  0)  and  y  =  30°.  33.  a  =  4:,b  =  S. 

18.  (0,  3)  andy  =  135°.  34.  a^  —  6,b  =  2. 

19.  (0,  0)  andy  =  120°.  35.  a  =  —  3,b  =  —  3. 

20.  (2,  —  3)  and  y  ==  0°.  36.  a  =  5,b  =  —  o. 

21.  (2,  — 3)  and  y  =  90°.  37.  a  =  —  10,b  =  5. 

22.  b  =  2,  y  =  4:0°.  38.a  =  l,b  =  —  l. 

23.  b  =  o,  y  =  4o°.  39.   a  =  n,  b  =  —  n. 
/  24.  6  =  — 4,  y  =  45°.  40.  a  =  n,b  =  'in. 

25.  &  =  — 4,  y  =  30°.  41.  79  =  5,  a  =  45°. 

26.  ^-  =  -4,  y  =  0°.  42.  ^J  =  5,  a  =120°. 

27.  i  =  -  4,  y  =  60°.  43.  2i  =  5,a  =  240°. 

28.  ^;  =  —  4,  y  =  90°.  44.  J^  =  5,a  =  300°. 


THE    STRAIGHT    LINE.  41 

Write  the  equations  of  the  sides  of  a  triangle : 

45.  If  its  vertices  are  the  points  (2, 1),  (3,  —  2),  ( —4,-1). 

46.  If  its  vertices  are  the  points  (2, 3),  (4, — 5),  ( — 3,  — 6). 

47.  Form  the  equations  of  the  medians  of  the  triangle 
described  in  example  46. 

48.  The  vertices  of  a  quadrilateral  are  (0,  0),  (1,  5), 
(7,  0),  (4,  — 9).  Form  the  equations  of  its  sides,  and  also 
of  its  diagonals. ' 

Find  the  equation  of  a  straight  line,  given  : 

49.  a  =  7^,  y  =  30°.  '51.    j!>  =  6,  y  =  45°. 

50.  «  =  — 3,  (xi,  ?/i)  is  (2,  5).     52.  p  =  6,  y  =  135°. 

Reduce  the  following  equations  to  the  symmetrical 
form,  and  construct  each  by  its  intercepts : 

53.  3x  — 2^  +  11  =  0  and  ?/=7x  +  l. 

54.  3x  +  5^  — 13  =  0  and4a;  — y  — 2  =  0. 

55.  Reduce  Ax-{-  Bi/=C  to  the  symmetrical  form;  also 
y  =  mx-{-h.  What  are  the  values  of  a  and  b  in  terras  of 
A,  B,  C,  and  m? 

Reduce  the  following  equations  to  the  form  i/  =  mx  +  b, 
and  construct  each  by  its  slope,  and  intercept  on  the  axis 
of  y: 
-56.    y  +  13  =5x  and  ?/  +  19  =  7x. 

57.  3:c  +  ?/  +  2  =  0  and2y  =  3a'  +  6. 

58.  Reduce  Ax -\-  Bi/=  C  to  the  form  //  =  tnx  +  b  ;  also 

--}--  =  l.     What  are  the  values  of  7?i  and  b  in  terms  of 
a      b 

A,  B,  C,  and  a  ? 

59.  Find  the  vertices  of  the  triangle  whose  sides  are  the 
lines  2.r  +  9y  +  17  =  0.  //  =  7.r - 38,  2i/-x=  2. 


42  ANALYTIC    GEOMETRY. 

'  •  60.  Find  the  oqiuitioii  of  the  str;iiglit  line  passing  tlirongh 
the  origin  and  the  intersection  of  tlie  lines  '3x  — 1*//-[-4  =  0 
and  o.r-|-4y  =  5.  Also  find  the  distance  between  these 
two  points. 

61.  AVhat  is  the  equation  of  the  line  passing  through 
(•^'i;  ^i))  ^^^^  equally  inclined  to  the  two  axes  ? 

62.  Find  the  equations  of  the  diagonals  of  the  parallelo- 
gram formed  by  the  lines  x  =  a,  x  =  h,  y  =  c,  y^^d. 

63.  Show  that  the  lines  ^=2x  +  3,  ?/  =  'ox  +  4,  y  =  4.^  +  5 
all  pass  tlirough  one  point. 

Hint.  Find  the  intersection  of  two  of  tlie  lines,  and  then  see  if  its 
coordinates  will  satisfy  the  equation  of  the  remaining  line. 

64.  The  vertices  of  a  triangle  are  (0,  0),  (xi,  0),  (.rg,  y^). 
Find  the  equations  of  its  medians,  and  prove  tliat  they 
meet  in  one  point. 

7  65.  What  must  be  the  value  of  in  if  the  line  y  =  mx 
passes  through  the  point  (1.  4)? 

r  66.  The  line  y  =  mx  +  3  passes  through  the  intersection 
of  the  lines  y  =^  x  -\- 1  and  y  =  2x-  -|-  2.  Determine  the 
value  of  /«. 

67.  Find  the  value  of  h  if  the  line  v/  =  Ox  +  ^  passes 
through  the  point  (2,  3). 

68.  What  condition  must  be  satisfied  if  the  points 
(•^1)  yi);  (^2)  Z/2)  (^3?  ys)  lie  in  one  straight  line? 

Hint.  Let  equation  [4]  represent  the  line  through  (xi,  ?/i)  and 
(-'^2,  2/2);  then  (X3,  2/3)  must  satisfy  it. 

69.  Discuss  equation  [5]  for  the  following  cases :  (i) 
(xi,  yi)  is  (0,  0),  (ii)  m  =  0,  (iii)  ni  =  cc. 

70.  Discuss  equation  [6]  for  the  following  cases:  (i) 
h  =  0,  (ii)  m  =  0,  (iii)  7n  =  00,  (iv)  m  =  0,  and  b^=0. 

71.  Discuss  equation  [7]  for  the  following  cases:  (i) 
a^^h,  (ii)  rt^O,  (iii)  a  =  x,  (iv)  b  =  co. 


the  stkaioht  link.  43 

General  Equation  of  tijk   First  Dkguee. 
42.   If  any  -point  P(xi,  7/1)  is  connected  with  the  origin  0  ; 


then  -^  =  cos  XOP,  i^~  =  sin  XOF,  and  OP  =  \/x{'  +  yf . 
OF  OF  i    I  ji 

Hence,  if  an//  two  real  quantities  are  each  divided  by  the 
square  root  of  the  sum,  of  their  squares,  the  quotients  are  the 
cosine  and  sine  of  some  angle. 

43.  The  locus  of  every  equation  of  the  first  degree  in  x 
and  y  is  a  strciight  line. 

Any  simple  equation  in  ,r  and  //  can  be  reduced  to  the 
form 

Aoc+By=C,  [9] 

in  which  C  is  positive  or  zero. 

Dividing  both  members  of  [9]  by  yj A'-\-  r>'-^,  we  obtain 

^  ^  ^'  /1^ 


V^'+5'      V^P+7>'''      V^-'+/^' 

Now,  by  §  42,  the  coefficients  of  x  and  y  in  (1)  are  a  set  of 
values  of  cos  a  and  sin  a,  and  the  second  member,  being  posi- 
tive, is  some  value  of  ^;  (§41).  Hence  (1)  is  in  the  normal 
form,  and  its  locus  is  some  straight  line.  Whence  the 
proposition. 

Cor.  1.  To  reduce  any  simple  equation  to  the  normal 
form,  put  it  in  the  form  of  [9],  and  divide  both  members  by 
the  square  root  of  the  sum  of  the  squares  of  the  coefficients 
of  X  and  y. 

CoK.  2.  To  construct  (1),  locate  the  point  '{A,  B),  con- 
nect it  with  the  origin,  and  on  this  line  lay  off  OS  equal  to 
the  second  member  of  (1)  ;  the  perpendicular  to  OiS  through 
Sis  the  locus  of  (1).  or  [9]. 

44.  The  locus  of  an  cquntidu  of  the  first  degree  in  .t  and 
y  is  called  a  Locus  of  the  First  Order. 


44  ANALYTIC    GEOMETRY. 

Exercise  12. 

Reduce  the  following  equations  to  the  normal  form,  and 
thus  determine  p,  or  the  distance  of  eacli  locus  from  the 
origin  : 

1.  3x-  — 2^+11  =  0.  5.    y-\-VS  =  5x. 

2.  3^3  +  5^  —  13  =  0.  6.    y+ 19  =  70:. 

3.  4x— y— 2  =  0.  7.    e.r  +  ry  +  ?i  =  0. 

4.  2x-{-o//=7.  8.    7i//-\-cx  —  r^^O. 
Eeduce  the  following  equations  to  one  of  the  forms  [6], 

[7],  [8],  and  determine  by  the  signs  of  the  constants  which 
of  the  four  quadrants  each  lodus  crosses  : 

9.    y  =  ^x  —  9.  14.    5a;4-4y  — 20  =  0. 

10.  3x-+2  =  2y.  15.    i/  =  6x-\- 12. 

11.  4.y  =  o.r  — 1.  -^16.    )/-\-2  =  x—4:. 

12.  4?/  =  3a: +  24.  17.    a;  +  V%  +  10  =  0. 

13.  5x  +  3?/  +  15  =  0.  18.   a;— VSy  — 10  =  0. 

19.  Discuss  equation  [9]  for  the  following  cases  : 

(i)   ^  =  0.      (iv)    A  =  cc.  (vii)     A  =  B,C=0. 

(ii)    i?  =  0.       (v)    A=C  =  0.      (viii)     A  =  -B,C  =  0. 
(iii)    C  =  0.      (vi)    A  =  B. 

20.  Reduce  equation  [7]  to  the  form  of  equation  [6], 
and  find  the  value  of  m  in  terms  of  a  and  b. 

21.  What  must  be  the  value  of  C  that  the  line  4x  —  5y  =  C 
may  pass  through  the  origin  ?   through  (2,  0)  ? 

22.  Determine  the  values  of  A,  B,  and  C,  that  the  line 
Ax-\-  By=C  may  pass  through  (3,  0)  and  (0,  — 12). 

Hint.  Since  the  coordinates  of  the  given  points  must  satisfy  the 
equation,  we  have  the  two  relations  ZA  =  C  and  —  12B  =  C. 

23.  From  [9]  deduce  [4]  by  the  method  used  in  No.  22. 


THE    STRAIGHT    LINE. 


45 


24.  If  equations  [4]  and  [9]  represent  the  same  line, 
what  are  the  values  of  A,  B,  C,  in  terms  of  Xi,  i/i,  x^,  y-i"^ 

25.  In  equation  [4]  find  the  values  of  vi  and  b  in  terms 

of  Xi,    iji,   X2,    1/2- 

Angles. 

45.  To  find  the  angle  formed  by  the  lines  y  ^  mx -\- b, 
and  y  =  m'x  -\-  b'. 

Let  AB  and  CD  (Fig.  21)  represent  the  two  lines  respec- 
tively, meeting  in  the  point  P. 

Let  the  angle  AFC^^cf) ;  then,  by  Geometry,  <^  =  y  — y'. 

Whence,  by  Trigonometry, 

7n  —  in' 


tan  <t>  =  -,    , 

1  +  mm 

This  equation  determines  the  value  of  <^ 

Y 


[10] 


Fig.  21. 


Fig.  22. 


Cor.  1.  If  the  lines  are  parallel,  tan  <^= 0 ;  hence,  7n=  m'. 
Conversely,  if  m^=m',  <^  =  0,  and  the  lines  are  parallel. 
CoR.  2.  If  the  lines  are  perpendicular,  tan  <^=x  ;  hence, 

1  4- m )ii'  =  0,  or  ni'  — .       Conversely,  if    1  +  m m'  =  0, 

m 

<^:=90°,  and  the  lines  are  perpendicular. 


46  ANALYTIC    GEOMETRY. 

46.  To  find  the  equatioti  of  a  strnujlit  line  jjcissiiif/  throiujk 
the  ijoint  (a'l,  y/i)  and  (i)  'parallel,  (ii)  ijerjiendicMlat',  to  the 
line  y  =  mx  -f-  i>- 

The  slope  of  the  required  line  is  m  in  case  (i),  and ■ 

in  case  (ii)  ;  and  in  both  cases  the  line  passes  through  a 
given  point  (xi,  y^). 

Therefore  (§  38),  the  required  equation  is 

(i)  !/  —  I/i  =  '>^K^—'-^i), 

i3xercise  13. 

Find  the  equation  of  tlie  straight  line  : 

1.  Passing  through  (3,  —  7),  and  ||  to  the  line  y  =  8x  —  5. 

2.  Passing  through  (5,  3),  and  ||  to  the  line  ^y  —  ^x=l. 

3.  Passing  through  (0,  0),  and  ||  to  the  line //  —  4x^10. 

4.  Passing  through  (5,  8),  and  ||  to  the  axis  of  x. 

5.  Passing  througli  (5,  8),  and  ||  to  the  axis  of  y. 

6.  Passing  through  (3,  —13),  and  _L  to  the  line  y=4:X  —  7. 

7.  Passing  through  (2,  9),  and  _L  to  the  line  7y-\-23x 
-5  =  0. 

8.  Passing  through  (0,  0),  and  _L  to  the  line  x-\-2y=^l. 

9.  Perpendicular  to  the  line  5x  —  7y  +  1  =  0,  and  erected 
at  the  point  whose  abscissa  =  1. 

47.  To  find  the  eqiiation  of  a  straight  line  passing 
through  a  given  point  (xi,  y^,  and  malting  a  given  angle  ^ 
with  a  given  line  y  =  mx  -\-  b. 

Let  the  required  equation  be 

y  —  !Ji  =  '":^'{^  — ^i)j 
where  in'  is  not  yet  determined. 


THE    STRAIGHT    LINK.  47 

Since  the   required  line   may  lie  eiciier  as    PQ  or  I'R 
(Fig.  22),  we  shall  liave  (§45), 

m'  —  m         VI  —  m' 
tan  <^  =  -T— ^  or 


1  +  fti^^'       1  +  tnm' 

,       m  ±  tan  <b 

Hence,  m'  =  - ^— > 

1  ip  ni  tan  ^ 

and  the  required  equation  is 

tn  ±  tail  <j)  ^ .  .  ^ 

2/  -  2/1  =  7^„,  tan  4.  ^'^  "  '^i)'  C^l] 

and  (as  Fig.  22  sliows)  there  are  in  general  two  straight 
lines  satisfying  the  given  conditions. 

Exercise  14. 

1.  Find  the  angle  formed  by  the  lines  .r-f-2//  +  t=0 
and  x  —  3]/  —  4^0. 

The  two  slopes  are  —  -}  and  ?,.  If  we  put  m.—  —  ^,  in'  =  ?.,  we 
obtain  tan  </>  =  —  1,  <p  =  l.*]5°.  If  we  put  ni  =  J,  m'  =  —  i,  we  get 
tan  <p  =  I,  (f)  =  45°.     Sliow  that  both  these  results  are  correct. 

Find  the  tangent  of  the  angle  formed  by  the  lines  : 
,  ' 2.    3x  —  4:1/ =  7  and  2x  —  //  =  3. 

3.  2.r  +  .3//  +  4=0  and  Ih-  +  4//  +  .^>  =  0. 

4.  y  —  nx  =  1  and  2  (//  —  1)  ^  "•^• 
Find  the  angle  formed  by  the  lines : 

5 .  .r  -f-  ?/  =  1  and  ?/  =  ,T  +  4. 

6 .  .V  +  3  =  2x  and  ?/  +  3,r  =  2. 

7.  2x  +  3//  +  7  =  0  and  3.r  —  2//  +  4=0. 

8.  G.r  =  2//  +  r>  and  //  —  3.r  =  1 0. 

9.  X  +  3  =  0  and  y  —  V3.r  +  4  =  0. 


48  ANALYTIC    GEOMETRY. 

10.  Discuss  equation  [11]  for  the  cases  when  <^  =  0* 
and  <^  =  90°. 

Note.  The  learner  should  solve  the  next  four  exercises  directly 
without  using  equation  [11] ;  then  verify  the  result  by  means  of  [11]. 

Find  the  equation  of  a  straight  line  : 

11.  Passing  through  the  point  (3,  5),  and  maliing  the 
angle  45°  with  the  line  2x  —  3y  +  5  =  0. 

12.  Passing  through  the  point  ( — 2, 1),  and  making  the 
angle  45°  with  the  line  2//  =  6  —  3x. 

13.  Passing  through  that  point  of  the  line  y^=2x  —  l  for 
which  x  =  2,  and  making  the  angle  30°  with  the  same  line. 

14.  Passing  through  (1,  3),  and  making  the  angle  30° 
with  the  line  x  —  2//  +  1  =  0- 

•  15.    Prove  that  tlie  lines  represented  by  the  equations 

Ax  +  Bi/-{-C  =  0,     A'x  +  B'i/-\-C'  =  0 
are  parallel  if  AB'=^A'B ;  i:>erpendicular,  if  AA'^  —  BB'. 

y  16.  Given  the  equation  3aj  +  4y-|- 6  =  0;  show  that  the 
general  equations  representing  (i)  all  parallels  and  (ii)  all 
perpendiculars  to  the  given  line  are 

(i)     3.r  +  4y  +  7i  =  0. 
(ii)     4x  — 3y  +  /i  =  0. 

17.  Deduce  the  following  equations  for  lines  passing 
through  (xi,  ?/i)  and  (i)  parallel,  (ii)  perpendicular,  to  the 
line  y  =  vix  +  b. 

(i)     y  —  mx^  ?/i  —  mxi. 
(ii)     my  +  x  =  my^  +  x^. 

18.  Write  the  equations  of  three  lines  parallel,  and 
three  lines  perpendicular,  to  the  line  2a; +  3^ +  1=^0. 


THE    STRAIGHT  LINE.  49 

y    19.    Among  the  following  lines  select  parallel  lines  ;  per- 
pendicular lines  ;  lines  neither  parallel  nor  perpendicular : 

(i)  2a;  +  3?/ —  1  =  0.  (v)a;-?/  =  2. 

(ii)  ?>x  —  2ii  =  20.  (vi)  5  (a;  +  ?/)  -  1 1  =  0. 

(iii)  ^x-\-^y  =  0.  (vii)  cc  =  8. 

(iv)  12x  =  8^  -I-  7.  (viii)  7/  +  10  =  0. 

p^  20.    Prove  that  the  angle  <^,  between  the  lines 
Ja;  +  %  +  C'  =  0  and  A'x  +  BUj  +  C'=  0, 
is  determined  by  the  equation 

,      A'B  —  AB' 

21.    From  the  preceding  equation  deduce  the  conditions 
of  parallel  lines  and  perpendicular  lines  given  in  Xo.  15. 

Find  the  equation  of  the  straight  line  : 

V22.  Parallel  to  2x  -f-3^  +  6  =0,  and  passing  through  (5, 7). 
^23.    Parallel  to  2x-{- 1/  —  1  =  0,  and  passing  through  the 

intersection  of  3x  +  2^  —  59  ^  0  and  5x  —  7v/  +  6  =  0. 
y^  24.    Parallel  to  the  line  joining  ( — 2,  7)  and  (—4,  —  5), 

and  passing  through  (5,  3). 

25.  Parallel  to  y  =  mx-\-b,  and  at  a  distance  d  from  the 
origin. 

26.  Perpendicular  to  Ax-\- Bi/-{- C=^0,  and  cutting  an 
intercept  b  on  the  axis  of  i/. 

X      ■?/ 

27.  Perpendicular  to  -  +  j-  =  1,  and  passing  through  (a,  b). 

28.  Making  the  angle  45°  with  --\-j  =  l,  and  passing 
through  (a,  0). 

29.  Show  that  the  triangle  whose  vertices  are  the  points 
(2,  1),  (3,  —  2),  (—  4,  —  1)  is  a  right  triangle. 


50  ANALYTIC    GEOMKTRY. 

30.  The  vertices  of  a  triangle  are  ( — 1,  —  1),  ( — 3,  5), 
(7,  11).  Find  the  equations  of  its  altitudes.  ^*rove  that 
the  altitudes  meet  in  one  point. 

y^  31.  Find  the  equation  of  the  perpendicular  erected  at 
the  middle  point  of  the  line  joining  (5,  2)  to  the  intersec- 
tion of  a-  +  2y  — 11  =  0  and  9a;  —  2y  —  59  =  0. 

32.    Find  the  equations  of  the  perpendiculars  erected  at 
the  middle  points  of  the  sides  of  the  triangle  whose  vertices 
are  (5,-7),  (1,  11),  (— -1,  13).     Prove  that  these  perpen- 
diculars meet  in  one  point. 
-^3.    The  equations  of  the  sides  of  a  triangle  are 

x-\-i/-\-l  =  0,  3.r  +  5^  +  ll  =  0,  cr  +  2^  +  4  =  0. 
Find   (i)  the   equations    of  the  perpendiculars  erected  at 
the  middle  points  of  the  sides  ;  (ii)  the  coordinates  of  their 
common  point  of  intersection ;    (iii)  the   distance  of  this 
point  from  the  vertices  of  the  triangle. 

34.  Show  that  the  straight  line  passing  through  («,  h) 
and  (c,  d)  is  perpendicular  to  the  straight  line  passing 
through  (b,  —  a)  and  ((1,  —  c). 

35.  What  is  the  equation  of  the  straight  line  passing 
through  (iPi,  ?/i),  and  making  an  angle  ^  witli  the  line 
Ax  +  Bij+C^O? 

Distances. 

48.  Find  the  distance  from  the  point  ( — 4,  1)  to  the  line 
3a;  -  4y  +  1  =  0.  .  Ans.  3. 

The  required  distance  iS  the  length  of  the  perpendicular 
from  the  given  point  to  the  given  line.  The  method  that 
first  suggests  itself  is  to  form  the  equation  of  this  perpen- 
dicular, find  its  intersection  with  the  given  line,  and  com- 
pute the  distance  from  this  intersection  to  the  given  point. 

Let  this  metliod  be  followed  in  solving  the  above  problem 
and  the  first  iiv(;  pro1)Ienis  of  Exercise  15. 


THE    STKAIGHT    LINE.  51 

49.    To  find  tlte  distance  from  the  point  (rt\,  //i)  to  tJia  Hue 

X  cos  a 4".'/  siu  a^=j). 
Let  the  line         a  cos  a  +  // sin  a=:y/,  (1) 

which  is  evidently  parallel  to  the  given  line,  pass  tlirough 
the  given  point  (oi\,  y/j);  then  we  have 

Xi  cos  a  +  i/y  sin  a  =y>'. 
Therefore,  x^  cos  a  +  .'/i  sin  a — p^p — p- 

But  p'' — p  equals  numerically  the  required  distance. 
Tliereforc,  the  distance  from  the  point  (a-,,  ij-^  to  tlie  line 
X  cos  a-\-  ij  sin  a  =7>  is  obtained  by  substituting  Xx  for  x  and 
2/1  for  y  in  the  expression  x  cos  u  +  //  sin  a  — p. 

CoR.  1.  The  distance  as  obtained  from  the  formula  will 
evidently  be  positice  or  negatwe  according  as  the  point  and 
origin  are  on  o})posite  sides  of  the  line,  or  on  the  same  side. 
CoR.  2.  If  the  equation  of  the  line  is 
xix  -\-  Bij  =  C, 
and  d  denotes  the  distance  from  (.r,,  ij{)  to  the  line ;  then, 
evidently, 

Axx  -t  B\)x  —  €  p..,-. 

Hence,  to  find  the  distance  from  the  point  (x^,  y^)  to  the 
line  Ax-\-B}j—  C,  write  x^  for  x,  and  iji  for  y  in  the  ex- 
pression  Ax-\-  By—  C,  and  divide  the  residt  by  ^A--\-  />'-. 

For  example,  let  (—1,  3)  be  the  point,  and  2x-\-4:  =  'Sy 
the  equation  of  the  line. 

Putting  this  equation  in  tlie  form  of  [9],  we  have 

-2x-j-3y  =  4. 

-2  (-1)4-3x3-4        ,  ,_ 

Whence,  d  = ^pf^l^^^ =  +  /rr  Vl3. 

Hence,  the  point  and  origin  are  on  opposite  sides  of  the  line. 
If  only  the  length  of  d  is  sought,  its  sign  nuiy  be  neglected. 


52  ANALYTIC    GEOMETRY. 

Exercise  15. 

-    1.    Find  the  distance  from  (1, 13)  to  the  line  Sx=^y  —  5. 

2.  Find  the  distance  from  (8,  4)  to  the  line  y^=1x  — 16. 

3.  Find  the  distance  from  the  origin  to  the  line 
3x-  +  4y  =  20. 

4 .  Find  the  distance  from  (2, 3)  to  the  line  2x-\-y  —  A.  =0, 

5.  Find  the  distance  from  (3,  3)  to  the  line  y=^^x  —  9. 

6.  Prove  that  the  distance  from  the  point  (x^,  y^)  to  the 

line  y  =  mx  -(-  &  is 

v/i  —  mxi  —  h 

d^^'- ,  — > 

VI +m^ 

the  upper  or  lower  sign  being  used  according  as  b  is  posi- 
tive or  negative.  Express  this  result  in  the  form  of  a  rule 
for  practice. 

7.  Find  the  distances  from  the  line  3a:  -f  4?/  + 15  ^  0  to 
the  following  points:  (3,  0),  (3,-1),  (3,-2),  (3  —  3), 
(3,  -  4),  (3,  -  5),  (3,  -  6),  (3,  -  7),  (0,  0),  (- 1,  0),  (-  2, 0), 
(-3,0),  (-4,0),  (-5,0),  (-6,0). 

8.  Find  the  distances  from  (1,  3)  to  the  following  lines: 

3a;4-4y  +  15  =  0.  3a;  +  4?/—    5  =  0. 

3a; +  4//+ 10  =  0.  3a; +  4^—10  =  0. 

3a;  +  4y+    5  =  0.  Sx-\- ^y  —  15  =  0. 

3x^Ay  =0.  3x  +  4?/  — 20  =  0. 

Find  the  following  distances: 

9.  From  the  point  (2,  —  5)  to  the  line  y  —  3x  =  7. 

10.  From  the  point  (4,  5)  to  the  line  4:y-\-5x  =  20. 

11.  From  the  point  (2,  3)  to  the  line  x-\-y^l. 
.12.    From  the  point  (0,  1)  to  the  line  Sx  —  3y  =  l. 

13.    From  the  point  (—1,  3)  to  the  line  Sx -{-4y-\-2  =  0. 


THE    STRAIGHT    LINE.  53 

-/  14.    From  the  origin  to  the  line  ?>x-\-2i/  —  G^O. 

15.  From  the  point  (2,  —  7)  to  the  line  joining  (—  4,  1) 
and  (3,  2). 

16.  From  the  line  y^=lx  to  the  intersection  of  the  lines 
y  =  3x  —  4  and  y=^ox-\-2. 

17.  From  the  origin  to  the  line  a(x  —  a)  +  b()/  —  ^)  =  0. 

18.  From  the  points  (a,  h)  and  ( — a,  —b)  to  the  line 

^+1=1. 

a      b 

19.  From  the  point  {a,  b)  to  the  line  ax  -\-bij=^0. 

20.  From  the  point  (Ji,  k)  to  the  line  Ax-\-  Bf/-\-  C  =  D. 

Find  the  distance  between  the  two  parallels : 

21.  3x'  +  4^  +  lo  =  0  and  3a:  +  4y  +  5  =  0. 
^22.    3a;  +  4y  +  lo  =  0  and  3.>;  +  4y  — 5  =  0. 

23.  Ax-\-Bt/=C  and  Ax-\-B/j=C'. 

24.  Ax  +  Bi/=Caud—Ax  —  Bi/=C'. 

25.  y  =  5x  —  7  and  y  =  5x  -\-  3. 

26.  -  +  f  =  2  and  -  +  7  =  0- 
a      b  a       b       Z 

27.  Show  that  the  locus  of  a  point  which  is  equidistant 
from  the  lines  Sx  +  4y  — 12  =0  and  4.r  +  3//  —  24  =  0  con- 
sists of  two  straight  lines.  Find  their  equations,  and  draw 
a  figure  representing  the  four  lines. 

28.  Show  that  the  locus  of  a  point  which  so  moves  that 
the  sum  of  its  distances  from  two  given  straight  lines  is 
constant  is  a  straight  line. 


54 


ANALYTIC    GEOMETRY. 


r 

/ 

\ 

/ 

/ 

\ 

> 

^r 

0 

L 

I 

v" 

~1 

IX 

Areas. 

50.     To  find  fhe  area  of  a  triangle,  having  given  its  vertices. 

Solution  I.  Let  PQR  (Fig.  23)  be  the  given  triangle, 
and  let  the  vertices  of  PQR  be  (x^,  y{),  (x^,  y^,  {x^,  y^,  re- 
spectively.    Drop  the  perpendiculars  PM,  QN,  RL;  then 

area  PQR  =  PQNM+  RLNQ  —  PMLR. 

By  Geometry, 

PQNM=  ^  NM{MP  -\-  NQ) 

^^(a-i  — 3-2)(//i  +  y2). 

Similarly, 

RL]S'Q  =  ^{x^-x,)iy,  +  y.^. 

PMLR  =  ^{x,-x,)(y,  +  y,). 
Substituting  these  values,  we  have 

Fig  23.  ° 

area  PQR  =  ^  \_{j\  —  x^)  {y.  +  ?/i) 

+  (.^2  —  x^){yz  +  .^2)  —  (^1  —  ^?)(!h  +  //i)] 
=  i  [—  -'"i.'/i  +  ^i//2  ~  -"^zlh  +  -^'"J/i  ~  •'•1//3  +  ■'•;;//!]• 
.-.  area  =  i  [ici  (2/2  -  */a)  +  x-i  (j/3  -!/i)+  .^s  (yi-2/2)].      [13] 
Solution  II.    Since  the  area  of  a  triangle  is  equal  to 
one  half  the  product  of  its  base  and  its  altitude,  this  prob- 
lem may  be  solved  as  follows  : 

(i)  Find  the  length  of  any  side  as  base, 

(ii)  Find  the  equation  of  the  base. 

(iii)  Find  the  distance  of  the  base  from  the  opposite  vertex, 

(iv)  Multii)ly  this  distance  by  one  half  the  base. 

Exercise  16. 
Find  the  area  of  the  triangle  whose  vertices  are  the  points : 

1.  (0,0),  (1,2),  (2,1).  3.    (2,3),(4,-5),(-3,-G). 

2.  (3, 4),  (-3, -4),  (0,4).      4.    (8,  3),  (-2, 3),  (4, -5). 

5.    (a,0),  (-«,0),  (0,^.). 


THE    STKAIGIIT    LINE. 


6.  Compare  the  ionuula  for  the  area  of  a  triangle  with 
the  result  obtained  by  solving  No.  68,  p.  42.  What,  then, 
is  the  geometric  meaning  of  that  result? 

Find  the  area  of  the  figure  having  for  vertices  the  points  : 

7.  (3,  5),  (7,  11),  (9,  1). 

8.  (3, -2),  (5,  4),  (-7,  3). 

9.  (-1,2),  (4,  4),  (6, -3). 

10.  (0,0),  (x,,!/,),  (x„,!/,). 

11.  (2,-r,),  (2,8),  (-2,-5). 

12.  (10,5),  (-2,5),  (-5,-3),  (7,-3). 

13.  (0,  0),  (5,  0),  (9, 11),  (0,  3). 

14.  (a,  1),  (0,  b),  (c,  1). 

15.  (a,  b),  {h,  a),  (c,  c). 

16.  {a,h),  {h,a),  {c,  —  <'). 

17.  Find  tlie  angles  and  the  area  of  the  triangle  whose 
vertices  are  (3,  0),  (0,  3  V3),  (G,  3  V3). 

What  is  the  area  contained  by  the  lines  : 

18.  a:  =  0,     y  =  0,     5j-1-4//  =  20? 

19.  x-\-l/=l,      .r  — //=(),      v/=0? 

20.  a;  +  2//  =  5,      2.r  +  //  =  7,      y  =  x-\-l? 

21.  a'  +  y  =  0,     .r  =  //,      //  =  3ft  ? 

22.  //  =  3a-,      If  =  7.r,      //  ^  r-  ? 

23.  a-  =  0,      >j  =  0,      j--4  =  0,      //  +  (;=0? 

24.  3x  +  //  4-  4  =  0,  3.r  -  5//  +  34  =  0,  3.i-  -  2//  +1=0? 

25.  X- 5// +  13  =  0,    5.r  +  7// +  1  =  0.    3.r  +  //-9  =  0? 

26.  X  —  I/  —  0,     X  +  //  =  0,      X—  i/  —  (f,     X  -\-  //  =  b  ? 


50  ANALYTIC    GEOMETRY. 

Find  the  area  contained  by  the  lines: 

27.  x  =  0,     y  =  0,     y^:^mx-\-b. 

28.  a;  =  0,     tj^Q,     -  +  '{  =  1. 

29.  x  =  0,     v/  =  0,     Ax-{-Bi/-\-C  =  0. 

30.  7/==  3a;  — 9,    y  —  Sx-'rS,    2i/  =  x  —  6,    2?/  =  a; +  14. 

31.  What  is  the  area  of  the  triangle  formed  by  drawing 
straight  lines  from  the  point  (2,  11)  to  the  points  in  the 
line  y=  5x  —  6  for  which  x  =  4:,  a:  =  7 ? 

Exercise  17.     (Revie\Ar.) 

1.  Deduce  equation  [7],  p.  39,  from  equation  [6]. 

2.  The  equation  y  =  mx  -\-b  \^  not  so  general  as  the 
equation  Ax-\-  By-\-  C=0,  because  it  cannot  represent  a 
line  parallel  to  the  axis  of  y.     Explain  more  fully. 

Determine  for  the  following  lines  the  values  of  a,  b,  y,  p,  and  a : 

3.  a;  =  2.  6.    x+\/3y  =  2. 

4.  x  =  7/.  7.    x  —  \/3y  =  2. 

5.  7/  +  l  =  V3(.x  +  2).  8.    V3x  — ^  =  2. 

9.  Find  the  equations  of  the  diagonals  of  the  figure  formed 
by  the  lines  3a-  —  y  +  9  =  0,     3x  =  y—l,     5a; +  3// ^18, 
5x  +  3y  =  2.     What  kind  of  quadrilateral  is  it  ?     Why  ? 

10.  Find  the  distance  between  the  parallels  9x=^y-\-l 
and  9a;  =  ?/  —  7. 

11.  The  vertices  of  a  quadrilateral  are  (3,  12),  (7,  9), 
(2,  —  3),  (—  2, 0).   Find  the  equations  of  its  sides  and  its  area. 

12.  The  vertices  of  a  quadrilateral  are  (6,  —  4),  (4,4), 
(—4,2),  ( — 8,-6).  Prove  that  the  lines  joining  the 
middle  points  of  adjacent  sides  form  a  parallelogram. 
Find  the  area  of  this  parallelogram. 


THE    STRAIGHT    LINE.  57 

Find  the  equation  of  a  line  passing  through  (3, 4),  and  also  : 

13.  Perpendicular  to  the  axis  of  x. 

14.  Making  the  angle  45°  with  the  axis  of  x. 

15.  Parallel  to  the  line  ox  +  6^  -}-  8  ==  0. 

16.  Intercepting  on  the  axis  of  y  the  distance  — 10.  " 

17.  Passing  through  the  point  halfway  between  (1,  —  4) 
and  (-  5,  4). 

18.  Perpendicular  to  the  line  joining  (3,  4)  and  (—1,0). 
Find  the  equations  of  the  following  lines: 

19.  A  line  parallel  to  the  line  joining  (xi,  y^)  and  (a-j,  ya)? 
and  passing  through  (ccj,  y^. 

20.  The  lines  passing  through  (8,  3),  (4,3),  (—5,-2). 

21.  A  line  passing  through  the  intersection  of  the  lines 
2ic  +  5y  +  8  =  0  and  2>x  —  4v/  —  7  =  0,  and  X  to  the  latter 
line. 

22.  A  line  JL  to  the  line  4x  —  ?/  =;  0,  and  passing  through 
that  point  of  the  given  line  whose  abscissa  is  2. 

23.  A  line  ||  to  the  line  3a:-|-4y  =  0,  and  passing 
through  the  intersection  of  the  lines  x  —  2y  —  a  =  0  and 
x-\-'6y  —  2a  =  ^. 

24.  A  line  through  (4,  3),  such  that  the  given  point  bi- 
sects the  portion  contained  between  the  axes. 

25.  A  line  through  (a-j,  ?/i),  such  that  the  given  point 
bisects  the  portion  contained  between  the  axes. 

26.  A  line  through  (4,  3),  and  forming  with  the  axes  in 
the  second  quadrant  a  triangle  whose  area  is  8. 

27.  A  line  through  (4,  3),  and  forming  with  the  axes  in 
the  fourth  quadrant  a  triangle  whose  area  is  8. 

28.  A  line  through  ( — 4,3),  sucli  that  the  jwrtion  be- 
tween the  axes  is  divided  by  the  given  point  in  the  ratio  5:3. 


58  ANALYTIC    GEOMETRY. 

29.  A  line  dividing  the  distance  between  ( —  3,  7)  and 
(5,  —  4)  in  the  ratio  4:7,  and  JL  to  the  line  joining  these 
points. 

30.  The  two  lines  through  (3,  5)  making  the  angle  45° 
with  the  line  2x  —  3i/  —  7=0. 

31.  The  two  lines  through  (7,  —  5)  that  make  the 
angle  45°  with  the  line  ijx  —  2i/-\-o  =  0. 

32.  The  line  making  the  angle  45°  with  the  line  joining 
(7,  —  1)  and  ( — 3,5),  and  intercepting  the  distance  5  on 
the  axis  of  x. 

33.  The  two  lines  that  pass  through  the  origin  and 
trisect  the  portion  of  the  line  x-\-  i/  =  l  included  between 
tlie  axes. 

34.  The  two  lines  ||  to  the  line  4^- -1-5^  + 11  =  0,  at  the 
distance  3  from  it. 

35.  The  bisectors  of  the  angles  contained  between  the 
lines  //  =  2x-{-4:  and  —  ]/^3x  -\-6. 

Hint.  Every  point  in  the  bisector  of  an  angle  is  equidistant  from 
tlie  sides  of  the  angle. 

36.  The  bisectors  of  the  angles  contained  between  the 
lines  2x  —  5y  ==  0  and  4a-  +  3y  =  12. 

37.  The  two  lines  that  pass  through  (3, 12),  and  whose 
distance  from  (7,  2)  is  equal  to  V58. 

38.  The  two  lines  that  pass  through  ( — 2,  5),  and  are 
each  equidistant  from  (3,  — 7)  and  (—4,  1). 

Find  the  angle  contained  between  the  lines: 

39.  //  +  3  =  2x  and  //  +  3.r  =  2. 

40.  >/  =  5x  —  7  and  5y -{-  Ji —  3  =  0. 


THE    STRAIGHT    JASK.  59 

Find  the  distance  : 

41.  From  the  intersection  of  the  lines  3x-\-2i/-\-  4  =  0, 
2x  +  5y  +  8  =  0  to  the  line  y  =  5x  +  6. 

X         II 

42.  From  the  point  (A,  Ic)  to  the  line  -4--=  1. 

ah 

43.  From  the  origin  to  the  line  hx  -\-  hy  =  c'^. 

44.  From  the  point  (a,  0)  to  the  line  y=:??u-|-  -• 

m 

Find  the  area  included  by  the  following  lines: 

45.  x  =  //,    X  -\-  y  =^  0,    X  =  r. 

46.  x-\-i/  =  k,    2x  =  y  +  /.-,  2//  —  X  -\-  k. 

47.  -  +  '^  =  1,    y  =  2x-\-h,    x=2ii-\-a. 
a       h  ^  ' 

48.  ?/=4.r  +  7    and  the  lines    that   join  the    origin  to 
those  points  of  the  given  line  whose  ordinates  are  — 1  and  19. 

49.  The  lines  joining  the  middle  points  of  the  sides  of  tlie 
triangle  formed  by  the  lines  a: — 5^+1 1  =0, 11a- -|-C// — 1  =<>. 

a;  +  ?/  +  4  =  0. 

50.  Find  the  area  of  the  quadrilateral  whose  vertices  are 
((VO),  (0,5),  (11,  9),  (7,0). 

51.  What  point  in  the  line  5.)- —  4//  — 28  =  0  is  equidis- 
tant from  the  points  (1,  5)  and  (7,  — 3)  ? 

52.  Prove  that  the  diagonals   of  a  square  are  perpen- 
dicular to  each  other. 

53.  Prove  that  the  line  joining  the  middle  points  of  two 
sides  of  a  triangle  is  parallel  to  the  third  side. 

54.  What   is    the    geometric   meaning  of   the   equation 
xy  =  0? 


60  ANALYTIC    GEOMETKY. 

55.  Show  that  the  three  points  (3a,  0),  (0,  'SO),  (a,  2U) 
are  in  a  straight  line. 

56.  Show  that  the  three  lines  5a; +  3//— 7  =  0,  3a;  —  Aij 
— 10  =  0,  and  x-\-2i/^0  meet  in  a  point. 

57.  What  must  be  the  value  of  a  in  order  that  tlie  three 
lines  3x  +  ?/  — 2  =  0,  2x—y  —  S=0,  and  aa;  +  2y  — 3  =  0 
may  meet  in  a  point  ? 

What  straight  lines  are  represented  by  the  equations  : 

58.  x^-\-{a  —  b)x  —  ah=^0? 

59.  xy-[-hx-\-(.uj-\-ah—^'^ 

60.  x~ij  =  xif  ? 

61.  14a;^  —  5a-^  —  y- =  0  ? 

In  the  following  exercises  prove  that  the  locus  of  the 
point  is  a  straight  line,  and  obtain  its  equation  : 

62.  The  locus  of  the  vertex  of  a  triangle  having  the  base 
and  the  area  constant. 

63.  The  locus  of  a  point  equidistant  from  the  points 
(a;i,  yi)  and  {x^,  y^. 

64.  The  locus  of  a  point  at  the  distance  d  from  the  line 
Ax-\-By^-C  =  ^. 

65.  The  locus  of  a  point  so  moving  that  the  sum  of  its 
distances  from  the  axes  shall  be  constant  and  equal  to  k. 

66.  The  locus  of  a  point  so  moving  that  the  sura  of  its 
distances  from  the  lines  Ax-\-By-\-  C=0,  A'x-\-B'y-\-  C"=0 
shall  be  constant  and  equal  to  k. 

67.  The  locus  of  tlie  vertex  of  a  triangle,  having  given 
the  base  and  the  difference  of  the  squares  of  the  other  sides. 


THE    STRAIGHT    LINE.  61 

SUPPLEMENTARY    PPtOPOSITIONS. 

Lines  passing  through  One  Point. 

51.  If  S=0,  S'  =  0  represent  the  equations  of  any  tivo 
loci  with  the  terms  all  transposed  to  the  left-hand  side,  and  k 
denotes  an  arbitrary  constant,  then  the  locus  represented  by 
the  equation  S-\-  kS'  =  0  pxisses  throuc/h  every  point  common 
to  the  two  given  loci. 

For  if  any  coordinates  satisfy  the  equation  *S'=0,  and 
also  satisfy  the  equation  *S'  =  0,  they  must  likewise  satisfy 
the  equation  S  +  kS^  =  0. 

For  what  values  of  k  will  the  equation  ^'-|- A-,S''  =  0  rep- 
resent the  lines  *S'  =  0  and  >S"  =  0,  respectively  ? 

52.  Find  the  equation  of  the  line  joining  the  point  (3,  4) 
to  the  intersection  of  the  lines 

3a?  — 2^+17=0  and  a; +  4?/  — 27  =  0. 

The  method  of  solving  this  question  that  first  suggests 
itself  is  to  find  the  intersection  of  the  given  lines  and 
then  apply  equation  [4],  p.  37. 

Another  method,  almost  equally  obvious,  is  to  employ 
equation  [5],  which  gives  at  once 

y  —  4  =  m(x  —  3), 

and  then  determine  m  by  substituting  for  x  and  y  the  coor- 
dinates of  the  intersection  of  the  given  lines. 

The  following  method,  founded  on  the  principle  stated 
in  §  51,  is,  however,  sometimes  preferable  on  account  of  its 
generality  and  because  it  saves  the  labor  of  solving  the 
given  equations.  According  to  this  principle,  the  required 
equation  may  be  immediately  written  in  the  form 

3x  -  2y  +  1 7  +  A-(.r  -f  Ay  -  27 )  -  0. 


62  ANALYTIC    OEOMETKY. 

And  since  the  line  passes  through  (3,  4j,  we  must  have 
9  -  8  +  17  +  /c(3  +  16  -  27)  =  0, 
whence,         k  =  '  • 

Therefore,  12^;  —  8//  +  08  -\-9x-\-  36i/  —  243  =  0, 
or  ox  +  4//  —  25  =  0. 

This  is  the  equation  of  the  required  line 

53.  .  If  the  equations  of  three  straight  lines  are 

Ax  +/.'//  -f-C  =0, 
A'x  ^  />"//  +  6"  =0, 
A"x  +/>'"//  +  C"'  ^0, 

and  we  can  find  three  constants,  I,  m,  n,  so  that  the  relation 
l{Ax-\- By-[-  C)^m{A'x+ir ij^C)-\-n{A"x-]- B" i/-^  C")^0 
is  identically  trne,  that  is,  true  for  all  values  of  x  and  y, 
then  the  three  lines  'meet  in  a  'point. 

For  if  the  coordinates  of  any  point  satisfy  any  two  of 
the  equations,  then  the  above  relation  shows  that  they  will 
also  satisfy  the  third  equation. 

54.  To  find  the  equation  of  the  bisector  of  the  angle 
between  the  two  lines 

X  cos  a  +  //  sin  a  =:  p, 
and  X  cos  a'  -\-  y  sin  a'  =^jj'. 

There  are  evidently  two  bisectors :  one  bisecting  the 
angle  in  which  the  origin  lies  ;  the  other  bisecting  the 
supplementary  angle. 

Now,  every  point  in  either  bisector  is  equally  distant 
from  the  sides  of  the  angle.  Let  (x,  y)  be  any  point  in  the 
bisector  of  the  angle  that  includes  the  origin  ;  then  (§  49) 

X  cos  a+ //sin  a— 7^:=  a- cos  a' 4-y  sin  a' — p\  (1) 

Since  (x,  y)  is  any  point  in  this  bisector,  (1)  is  its  equation. 


THE    STRAUJIIT    LINE.  63 

The  equation  of  the  other  bisector  is 

xcosa-{-  1/  sin  a — 2^  —  —  (;r  cos  a'  +  y  sin  a* — //).      (2) 

To  distinguish  equations  (1)  and  (2)  we  note  that  in  tlie 
first  the  constant  terms  in  the  two  members  have  like  signs  ; 
while  in  the  second  the  constant  terms  have  unlike  signs. 

CoR.  1.     If  the  equations  of  the  lines  are  in  the  form 
Ax  +Bi/  =  C,  A'x  +  B'l/  =  C, 
the  equations  of  the  bisectors  are  evidently 
Ax  +  Btj-  C  A'jr,  +  By  -  C 


V^2+^^  V^'2+"j5'i 


1[4] 


Equation  [14]  represents  the  bisector  of  the  angle  in 
which  the  origin  lies,  or  of  its  supplementary  angle,  accord- 
ing as  we  take  the  upper  or  lower  sign. 

For  example,  let  the  equations  of  the  lines  be 

2x  =  4?/ -f-  9,  and  5{/^=ox  —  7. 
Putting  these  equations  in  the  form  of  [9],  we  have 

2x  —  4//  =  9,  and  3x  —  5//  =  7. 
Hence,  the  equations  of  the  bisectors  of  tlieir  included 

angles  are      ^         .        ^  .,         ^        ^ 

2x  —  4y  —  9 ox  —  oy  —  / 

V20  ~         V34 

in  which  the  upper  sign  gives  the  equation  of  the  bisector 
of  the  angle  in  which  tlie  origin  lies. 

CoK.  2.  If  S=0  and  *S"=:0  represent  two  simple  equa- 
tions in  tlie  normal  form,  with  the  terms  all  transposed  to 
the  first  members,  then  the  equations  of  the  bisectors  of 
their  included  angles  may  be  written 

^=±^',  or  .S'^^'  =  0. 


64  AJTALYTIC    GEOMETRY. 

Exercise  18. 

Find  the  equation  of  the  line  passing  through  the  inter- 
section of  the  lines  3x  —  2y  +  1^  =  0,  x  +  4y  —  27  =  0,  and : 

1.  Passing  also  through  the  origin.  ' 

2.  Parallel  to  the  line  x  +  2i/-\-3  =  0. 

3.  Perpendicular  to  the  line  6x  —  5?/  =  0. 

4.  Equally  inclined  to  the  two  axes. 

5.  Find  the  equation  of   the  line  parallel  to  the  line 
x  =  y,  and  passing  through  the  intersection  of  the  lines 

y  =  2x-\-l  and  i/-\-3x^=  11. 

6.  Find  the  equation  of  the  straight  line  joining  (2, 3) 
to  the  intersection  of  the  lines 

2ic  +  3^+l=0  ■dnd3x  —  4i/  =  5. 

7.  Find  the  equation  of  the  straight  line  joining  (0,  0) 
to  the  intersection  of  the  lines 

5x  —  2i/-\-3=0  and  13x-\-2j  =  l. 

8.  Find  the  equation  of  the  straight  line  joining  (1, 11) 
to  the  intersection  of  the  lines 

2x-{-5i/  —  8  =  0  2ind3x  —  4>/  =  8. 

Find  the  equation  of  the  straight  line  passing  through 
the  intersection  of  the  lines  Ax  -\- Bi/  -\-  C^O  and  A'x-{- 
B't/-{-C'  =  0,  and  also: 

9.  Passing  through  the  origin. 

10.  Drawn  parallel  to  the  axis  of  x. 

11.  Passing  through  the  point  (xi,  t/i). 

12.  Find  the  equation  of  the  straight  line  passing  through 
the  intersection  of  5ar  — 4?/  +  3^0  and  7a;-[- lly— 1^0, 
and  cutting  on  the  axis  of  y  an  intercept  equal  to  6. 


THE    STRAIGHT    LINE.  65 

13.  Find  the  equation  of  the  straight  line  passing  through 
the  intersection  of  y^=-lx  —  4  and  y  =  —  2a3  +  5,  and  form- 
ing with  the  axis  of  "a;  the  angle  60°. 

14.  The  distance  of  a  straight  line  from  the  origin  is  5 ; 
and  it  passes  through  the  intersection  of  the  lines  3x  —  1y 
+  11  =  0  and  6x  +  7^  —  55  =  0.     What  is  its  equation  ? 

15.  What  is  the  equation  of  the  straight  line  passing 
through  the  intersection  of  bx-{-ay=^ab  and  y=mx,  and 
perpendicular  to  the  former  line  ? 

Prove  that  the  following  lines  are  concurrent  (or  pas?' 
through  one  point) : 

16.  ?/  =  2a-  +  l,     yz=x-{-3,     ?/=:  — 5x  +  15. 

17.  Ax—2y  —  ^  =  0,    3x-7/  +  ^  =  0,    5x  —  2y  —  l  =  0. 

18.  2x  —  y  =  5,     3cc  —  y  =  G,    4x  —  y  =  7- 

19.  What  is  the  value  of  77i  if  the  lines 

a      0  0      a 

meet  in  one  point  ? 

20.  When  do  the  straight  lines  y  =  mx  -\-b,y=^  vi'x  -\~  h', 
y^m"x-\-b"  pass  through  one  point? 

21.  Prove  that  the  three  altitudes  of  a  triangle  meet  in 
one  point. 

22.  Prove  that  the  perpendiculars  erected  at  the  middle 
points  of  the  sides  of  a  triangle  meet  in  one  point. 

23.  Prove  that  the  three  medians  of  a  triangle  meet  in 
one  point.  Show  also  that  this  point  is  one  of  the  two 
points  of  trisection  for  each  median. 

24.  Prove  that  the  bisectors  of  the  three  angles  of  a  tri- 
angle meet  in  one  point. 


06  ANALYTIC    GEOMETKY. 

25.  The  vertices  of  a  triangle  are  (2,1),  (3,-2),  (—4, 
—  1).  Find  the  lengths  of  its  altitudes.  Is  the  origin 
within  or  without  the  triangle? 

26.  The  equations  of  the  sides  of  a  triangle  are 

3j^  +  /y  +  4=0,    3.x- -5^  +  34=^0,    3a;  — 2^  +  l:=0. 
Find  the  lengtlis  of  its  altitudes. 

What  are  the  equations  of  the  lines  bisecting  the  angles 
between  the  lines  : 

27.  3x  —  4//  +  7  =  0  and  4.r  —  3y  + 17  =  0  ? 

28.  'Sx  +  4//  —  9  =  0  and  12x  +  5.7  —  3  =  0? 

29.  i/  =  2.r  —  4:and2t/  =  x-\-10? 

30.  x-\-f/  =  2  Sindx—>/  =  0? 

31.  y  =  mx  +  b  and  y  =  in'x  -\-  //? 

32.  Prove  that  the  bisectors  of  the  two  supplementary 
angles  formed  by  two  intersecting  lines  are  perpendicular 
to  each  other. 

Equations  representing  Straight  Lines. 

55.  A  homogeneous  eqvatwn  of  the  nth  degree  rejjresents 
n  straight  lines  through  the  origin. 

Let  the  equation  be 

j.r"  +  i;x"-'n  +  6'.r"  y  + +  Ay = 0. 

Dividing  by  .///",  we  have 

If  ^'i5  >'2>  '*3) '>'n  denote  the  roots  of  this  equation,  then 

the  equation,  resolved  into  its  factors,  becomes 


THE    STRAIGHT    LINE.  67 

and  therefore  is  satisfied  when  any  one  of  these  factors  is 
zero,  and  in  no  other  cases. 

Therefore,  the  locus  of  the  equation  consists  of  the  n 
straight  lines 

x  —  i\y  =  0,    x  —  r.,i/  =  0,    ,    x  —  r„y  =  0; 

and  these  lines  evidently  all  pass  through  the  origin. 

56.     To  find  the  angle  between  the  two  stniiyht  lines  rep- 
resented hxj  the  equation  Ax^  +  Cxy  -\-  Bif  =■  0. 

Solving  the  equation  as  a  quadratic  in  x,  we  obtain 
2 Ax  -I-  ((7  ±  ^C"-4AB)  t/  =  0. 

Hence,  the  slopes  of  the  two  lines  are 

2A  .  2A 


—  C-  VC2  -  4:AB     ■        —  C  +  VC'2 -  4:AB 
Therefore, 

,       VC'2— 4^^  ,      A 

VI  —  m  ^ ,   mm  =  —  ; 

B  B 

and  (equation  [10],  p.  45), 


m—m'       ^C^—AAB 
tan  (b  =  ~-, ,= — • 

57.    To  find  the  condition  that  the  general  emtation  of  the 
second  degree  may  represent  two  straight  lines. 

We  may  write  the  most  general  form  of  tlie  equation  of 
the  second  degree  as  follows  : 

Ax'  +  By-  +  Cxy  +  Bx  +  By  +  /<^  =  0.  (1 ) 

That  this  equation  may  represent  two  straight  lines,  its 
first  member  must  be  the  product  of  two  linear  factors  in 
X  and  y  ;  that  is,  the  equation  can  be  written  in  the  form 
(lx-]-7ny-\-n)  (px -\- qy-\-i')  =  0.  (2) 

Equating  coefficients  in  (1)  and  (2),  we  obtain 
Ij)  =  A,  niq  =  />,  n  r  =  F. 

Iq  -f-  nip  =  6',     //•  +  np  =  Z>,      nir  -\-  nq  =  E. 


68  AXALYTIC    GEOMETRY. 

The  product  of  C,  D,  and  E  is 

CDE  =  'llmvpqr  -\-  lp{ii?(f  -\-  m^i^)  +  mq(Pi^  +  n^p^y 

=  2ABF-\-  A  {E^  —  2BF)  +  B(I>'-  2AF) 
+  F{C^  —  2AB). 
Hence,  the  required  condition  is 

F{C^  —  AAB)  +  AE'  +  BD- -  CDE=  0.  (3) 

Exercise  19. 

1.  Describe  the  position  of  the  two  straight  lines  repre- 
sented by  the  equation  Ax^-{-Cxi/-\-  Bi/'-^-\-Dx-{-Ei/-\-F=0, 
when  (i)  A=C  =  I)  =  (),  (ii)  B=  C  =  E=0. 

2.  When  will  the  equation  ax>/-\-bx-\-ci/ -j-cl^O  repre- 
sent two  straight  lines  ? 

3.  Find  the  conditions  that  the  straight  lines  represented 
by  the  equation  Ax--\-  Cxy-\-  Bt/^^0  may  be  real  ;  imagi- 
nary ;  coincident ;  perpendicular  to  each  other. 

4.  Show  that  the  two  straight  lines  x^  —  2xy  sec  6+  ^^  =  0 
make  the  angle  6  with  each  other. 

Show  that  the  following  equations  represent  straight 
lines,  and  find  their   separate  equations  : 

6.  x''  —  2xy  —  ^if-\-2x  —  2y-]-l=0. 

7.  a;2  — 4a!y  +  3/  +  6?/— 9  =  0. 

8.  Show  that  the  equation  x^-\-xy  —  6^^  -\-lx-\-  Sly  — 
18  =  0  represents  two  straight  lines,  and  find  the  angle 
between  them. 

Determine  the  values  of  K  for  which  the  following  equa- 
tions will  re})resent  in  each  case  a  pair  of  straight  lines. 
Are  the  lines  real  or  imaginary  ? 

9.  12x^  —  10xy-\-2y^-{-llx  —  5y  +  K=0. 


THE    STRAIGHT    LIXE.  69 

10.  12x^  +  Kx>/-{-2if-\-llx  —  5i/-\-2  =  0. 

11.  12a;2  +  36a;?/  +  Ay  +  0./;+6y+3  =  0. 

12.  For  what  value  of  if  does  the  equation  Kxy-\-bx 
+  3y+2  =  0  represent  two  straight  lines? 

Problems  on  Loci  involving  Three  Variables. 

58.  A  trapezoid  is  formed  by  drawing  a  line  parallel  to 
the  base  of  a  given  triangle.  Find  the  locus  of  the  intersec- 
tion of  its  diagonals. 

If  ABC  is  the  given  triangle,  and  we  choose  for  axes  the 
base  AB  and  the  altitude  CO,  the  vertices  A,  B,  C  may  be 
represented  in  general  by  (a,  0),  (h,  0),  (0,  c),  respectively. 
The  equations  oi  AC  and  BC  are,  respectively, 

-  +  ^  =  land  ?  +  ^  =  l. 
a      c  be 

Let  y^m  be  the  equation  of  the  line  parallel  to  the  base, 
and  let  it  cut  ^C  in  D,  ^C  in  ^;  then  the  coordinates  of 
D  and  E,  respectively,  are 

f ac  —  am       \        ^    (be  —  biyi       \ 
(  J  m  1  and  (  >  m  I  • 

(1) 

X  —  a      be  —  bm  —  ac  ^  ' 

If  P  is  the  intersection  of  the  diagonals,  then  the  coordi- 
nates X  and  y  of  the  point  P  must  satisfy  both  (1)  and  (2) ; 
by  solving  these  equations,  therefore,  we  obtain  for  any 
particular  value  of  m  the  coordinates  of  the  point  P.  But 
what  we  want   is  the  algebraic  relation   that  is  satisfied 


Hence,  the  equation  of  the  diagonal  BD  is 

y    cm 

X  —  b      ac  —  am  —  be 
and  the  equation  of  the  diagonal  AE  is 
y  cm 


70  ANALYTIC    GKOMETllV. 

by  the  coordinates  of  P,  whatever  the  mine  of  m  may  be. 
To  find  this,  we  have  only  to  eliminate  m  from  equations 
(1)  and  (2).     By  doing  this  we  obtain 

2cx  +  {a  +  h)y  ={a-\-  b)c, 

X         ,   >/      . 
i(«  +  ^)      c 
We  see  from  the  form  of  this  equation  that  the  required 
locus  is  the  line  that  joins  C  to  the  middle  point  of  AB. 

Remark.  The  above  solution  should  be  studied  till  it  is  understood. 
In  problems  on  loci  it  is  often  necessary  to  obtain  relations  that 
involve  not  only  the  x  and  y  of  a  point  of  the  locus  which  we  are  seek- 
ing, but  also  some  third  variable  (as  m  in  the  above  example). 

In  such  cases  we  must  obtain  two  equations  that  involve  x  and  y 
and  this  third  variable,  and  then  eliminate  the  third  variable  ;  tlie 
resulting  equation  will  be  the  equation  of  the  locus  required. 

Exercise  20. 

1.  Through  a  fixed  point  0  any  straight  line  is  drawn, 
meeting  two  given  parallel  straight  lines  in  P  and  Q ; 
through  P  and  Q  straight  lines  are  drawn  in  fixed  direc- 
tions, meeting  in  R.  Prove  that  the  locus  of  i2  is  a 
straight  line,  and  find  its  equation. 

2.  The  hypotenuse  of  a  right  triangle  slides  between  the 
axes  of  X  and  y,  its  ends  always  touching  the  axes.  Find 
the  locus  of  the  vertex  of  the  right  angle. 

3.  Given  two  fixed  points,  A  and  B,  one  on  each  of  the 
axes  ;  if  Z7and  Fare  two  variable  points,  one  on  each  axis, 
so  taken  that  0U-{-  0F=  OA-^OB,  find  tlie  locus  of  the 
intersection  of  A  V  and  B  U. 

4.  Find  the  locus  of  the  middle  points  of  the  rectangles 
that  may  be  inscribed  in  a  given  triangle. 

5.  If  PP\  QQ'  are  any  two  parallels  to  the  sides  of  a  given 
rectangle,  find  the  locus  of  the  intersection  of  P'Q  and  PQ'. 


CHAPTER  III. 


THE  CIRCLE. 


Equations  of  the  Circle. 

59.  Tlie  Circle  is  the  locus  of  a  point  which  moves  so  tliat 
its  distance  from  a  fixed  point  is  constant.  Tlie  fixed  point 
is  the  centre,  and  the  constant  distance  the  radius,  of  the 
circle. 

Note.  The  word  "circle,"  us  here  detined,  means  the  same  thing 
as  "circumference"  in  Elementary  Geometry.  This  is  the  usual 
meaning  of  "circle"  in  the  higher  branches  of  Mathematics. 

60.  To  find  the  equation  of  a  circle,  having  given  its 
centre  («,  b)  and  its  radius  r. 

E 


Fig.  24. 

Let  C  (Fig.  24)  be  the  centre,  and  P  any  point  (x,  y)  of 
the  circumference.     Then  it  is  only  necessary  to  express  by 
an  equation  the  fact  that  the  distance  from  P  to  C  is  con- 
stant, and  equal  to  r :  the  required  equation  evidently  is  (§  6) 
(X  -  aY  +(!/-  W  =  /•■-.  [15] 


72  ANALYTIC    GEOMETRY. 

If  we  draw  CR  ||  to  OX,  to  meet  the  ordinate  of  P, 
then  we  see  from  the  figure  that  the  legs  of  the  rt.  A  CPR 
are  CR  =  x  —  a,  PR  =  1/  —  b. 

If  the  origin  is  taken  at  the  centre,  then  a  =  b:=0,  and 
the  equation  of  the  circle  is 

ic2  +  2/2  =  r2.  [16] 

This  is  the  simplest  form  of  the  equation  of  a  circle,  and 
the  one  most  commonly  used. 

If  the  origin  is  taken  on  the  circle  at  the  point  A,  and 
the  diameter  AB  is  taken  as  the  axis  of  x,  then  the  centre 
will  be  the  point  (r,  0).  Writing  r  in  place  of  a,  and  0  in 
place  of  b  in  [15],  and  reducing,  we  obtain 

x"+  y^  =  2rx.  [17] 

Why  is  this  equation  without  any  constant  term? 

61.  The  locus  of  any  equation  of  the  second  degree  in  x 
and  y  in  which  the  term  in  xy  is  wanting  and  the  coefficients 
of  x^  and  y^  are  equal  is  a  circle. 

Any  such  equation  can  evidently  be  reduced  to  the  form 
x''-^if-\-2Dx-\-2Ey-{-F=0.  (1) 

Therefore, 

{x^  +  2Dx  +  2)2)  +  (^fJ^2Ey  +  U^)  =  D" -\- E^  —  F, 
or  (x-\-Dy+{y-\-Ey=(iy-\-E^  —  F).  (2) 

Now,  from  [15]  it  follows  that  the  locus  of  (2)  is  a  circle 
whose  centre  is  ( — i>,  —  E),  and  whose  radius  is 

\/l)-^  +  E^  —  F. 

Cor.  If  Z)2  +  ^2  >  F,  the  radius  is  real  and  the  circle 
is  readily  constructed.  If  D^  -\-  E^=^F,  the  radius  is  zero, 
and  the  locus  is  the  single  point  ( — D,  —  E).  If  D^-\-E'^ 
<  F,  the  radius  is  imaginary,  and  the  equation  represents 
no  real  locus. 


THE    CIRCLE.  73 

'62.    Ajit/  point  (h,  k)  is  without,  on,  or  within  the  circle 
x^-{-  y^:=  r^,  according  as  h^  -\-  k-  >,  ==,  or  <  r^. 

For  a  point  is  without,  on,  or  within  a  circle,  according 
as  its  distance  from  the  centre  >,  =,  or  <  the  radius. 

Exercise  21. 

Find  the  equation  of  the  circle,  taking  as  origin  : 

1.  The  point  B  (Fig.  24)  ;  and  BA  as  axis  of  x. 

2.  The  point  D  (Fig.  24)  ;  and  DE  as  axis  of  y. 

3.  The  point  E  (Fig.  24)  ;  and  ED  as  axis  of  y. 

Write  the  equations  of  the  following  circles: 

4.  Centre  (5,  —  3),  radius  10. 

5.  Centre  (0,  —  2),  radius  11. 

6.  Centre  (5,  0),  radius  5. 

7.  Centre  ( — 5,  0),  radius  5. 

8.  Centre  (2,  3),  diameter  10. 

9.  Centre  (Ji,  k),  radius  V/i^  +  H 

10.  Determine  the  centre  and  radius  of  the  circle 

ic^  +  Z  — 10^+12^+25  =  0. 
Here  (x  -  5)2  +  (y  +  6)2  =  36.     .  •.  a  =  5,  6  =  -  6,  r  =  6. 

Determine  the  centres  and  radii  of  the  following  circles: 

11.  x''-\-f/  —  2x  —  4.y  =  0.     17.  6x2  — 2y(7  — 3y)=0. 

12.  3a;2+3/— 5x— 7?/+l=0.  18.  x-+if  =  9k\ 

13.  x''-\-y^  —  Sx  =  0.  19.  (x  +  yy-  +  (x  —  yy-=8k-. 

14.  cc2  +  /+8.r  =  0.  20.  x--\-y-  =  a-  +  b^ 

15.  x^-{-y--8y  =  0.  21.  x^  +  y'  =  k{x -\- k). 

16.  x^-\-f-j-Sy  =  0.  22.  x'-\-y-  =  hx-\-ky. 


74  ANALYTIC    GEOMETRY. 

23.  When  are  the  circles  x-  -\-  if  +  />./•  -\-  Eij  -[-  6'  =  0  and 
^•-  +  //'  +  J^>'-'^  +  ^^'//  +  C'  =  0  concentric  ? 

24.  Wliat  is  the    geometric   meaning  of  the  equation 

{x-af-\-{ij-l>f  =  i'^ 

25.  Find  the  intercepts  of  the  circles 

(i)  x2  +  /-8.«-8y+    7  =  0, 

(ii)  a;-  +  /  — 8a;  — 8y+16  =  0, 

(iii)  x'  +  }j-  —  8ic  —  8y  +  20  =  0. 

Putting  2/  =  0  in  each  case,  we  Iiave  in  case  (i)  x-  —  8x  +  7  =  0, 
whence  x  =  1  and  7;  in  case  (ii)  x'^  —  8x  +  16=  0,  whence  x  =  4;  in 
case  (iii)  x-  —  8x  +  20  =  0,  whence  x  =  ±  V—  4. 

Putting  X  =  0  in  each  case,  we  obtain  for  y  values  identical  with  the 
above  values  of  x. 

The  geometric  meaning  of  these  results  is  as  follows: 

Circle  (i)  cuts  the  axis  of  x  in  the  points  (1,  0),  (7,  0),  and  the  axis 
of  y  in  the  points  (0,  1),  (0,  7). 

Circle  (ii)  touches  the  axis  of  x  at  (4,  0),  and  the  axis  of  y  at  (0,  4). 

Circle  (iii)  does  not  meet  the  axes  at  all. 

This  is  the  meaning  of  the  imaginary  values  of  x  and  y  in  case  (iii). 

If,  however,  we  wish  to  make  tlie  language  of  Geometry  conform 
more  exactly  to  that  of  Algebra,  then  in  this  case  we  should  say  that 
the  circle  meets  the  axes  in  imaginary  points.  This  form  of  statement, 
however,  must  be  understood  as  simply  another  way  of  saying  that 
the  circle  does  not  meet  the  axes. 

Find  the  centres,  radii,  and  intercepts  on  the  axes  of  the 
following  circles: 

26.  a-2  4-;//2_5;y  — 7y4-6  =  0. 

27 .  x"  +  /  —  12x  —  4?/  + 15  =  0. 

28.  a--  +  if  —  4 J-  —  8//  =  0. 

29.  x'  +  //  -  (;./•  +  4//  +  4  =  0. 

30.  x'  +  y-  +  Tlx  —  1 8y  +  57  =  0. 


THE    CIRCLE.  75 

31.  Under  what  conditions  will  the  circle  y^ -\-  if -\- Dx 
-\-  Eif-\-  C  =  0  (i)  touch  the  axis  of  j!  ?  (ii)  touch  the  axis 
of  1/  ?  (iii)  not  meet  the  axes  at  all  ? 

32.  Show  that  the  circle  x'^-{-ir -\-10x  — 10 i/i- 25  =  0 
touches  the  axes  and  lies  entirely  in  the  second  quadrant. 
Write  the  equation  so  that  it  shall  represent  the  same 
circle  touching  the  axes  and  lying  in  the  third  quadrant. 

33.  In  what  points  does  the  straight  line  3x-\-  u  =  25 
cut  the  circle  x-  -\-  if  =  65  ? 

34.  Find  the  points  common  to  the  loci  x'^-\-if=^^  and 
y  =  2x  — 4. 

35.  The  equation  of  a  chord  of  the  circle  a--+ ?/-  =  25  is 
y^2x-\- 11.     Find  the  length  of  the  chord. 

~  X         1/ 

36.  The  equation  of  a  chord  is  - -f- 'r  ^=  1  j  that  of  the 
circle  is  x^-\-  )f  =  't'^.     Find  the  length  of  the  chord. 

37.  Find  the  equation  of  a  line  passing  through  the  centre 
of  a;^  +  ^^  —  6a;  —  ^y  =  —  21  and  perpendicular  to  a;  +  2//  =  4. 

38.  Find  the  equation  of  that  chord  of  the  circle  x--\-  if- 
=  130  that  passes  through  the  point  for  wliich  the  abscissa 
is  9  and  the  ordinate  negative,  and  that  is  parallel  to  the 
straight  line  4.x  —  5//  —  7  =  0. 

39.  What  is  the  equation  of  the  chord  of  the  circle 
a;2_|_^-'_-277  tliat  passes  through  (3,  —  5)  and  is  bisected 
at  this  point  ? 

40.  Find  the  locus  of  the  centre  of  a  circle  passing 
through  the  points  (xi,  y^)  and  (j-.^,  //o). 

41.  What  is  the  locus  of  the  centres  of  all  the  circles 
that  pass  through  the  points  (5,  3)  and  ( — 7,  — 6)  ? 


76  ANALYTIC    GEOMETRY. 

Find  the  equation  of  tlie  circle  : 

42.  Passing  through  the  points  (4,  0),  (0,  4),  ((5,  4). 

43.  Passing  through  the  points  (0,  0),  (8,  0),  (0,-6). 

44.  Passing  through  the  points  (—  6,  —  1),  (0, 0),  (0,  —  1). 

45.  Passing  through  the  points  (0,  0),  (—8a,  0),  (0,  6a). 

46.  Passingthroughtliepoints(2,  — 3),(3,  — 4),(— 2,  — 1). 
•y  47.  Passing  through  the  points  (1,  2),  (1,  3),  (2,  5). 

48.  Passing  through  (10, 4)  and  (17,— 3),  and  radius  =  13. 

49.  Passing  through  (3,  6),  and  touching  the  axes. 

50.  Touching  each  axis  at  the  distance  4  from  the  origin. 

51.  Touching  each  axis  at  the  distance  a  from  the  origin. 

52.  Passing  through  the  origin,  and  cutting  the  lengths 
a,  h  from  the  axes. 

53.  Passing  through  (5,  6),  and  having  its  centre  at  the 
intersection  of  the  lines  y  =  lx  —  3,  4^/  —  3a;  =  13. 

:  54.    Passing  through  (10,  9)  and  (5,  2  — 3V6),  and  hav- 
ing its  centre  in  the  line  3x  —  2y — 17  =  0. 

55.  Passing  through  the  origin,  and  cutting  equal  lengths 
a  from  the  lines  x  =  y,  x-\-y  =  0. 

56.  Circumscribing  the  triangle  whose  sides  are  the  lines 
y  =  0,   y  =  mx-\-b,  -  +  ^=1. 

57.  Having  for  diameter  the  line  joining  (0, 0)  and  (x^,  y^). 

58.  Having  for  diameter  the  line  joining  (x^,  ?/i)and(iC2,  y^- 

59.  Having   for   diameter   the  line   joining  the  points 
where  y  ^=  nnx  meets  ic^  -|-  y/^  =  Irx. 

60.  Having    for   diameter   the    common    chord    of   the 
circles  x^  -\-y^^^  r^  and  (x  —  a)^  -\-y^  =  r\ 


THE    CIRCLE. 


77 


Tangents  and  jSTormals. 

63.  Let  QPQ'  (Fig.  25)  represent  any  curve.  If  the 
secant  QPR  is  turned  about  the  point  P  until  the  point  Q 
approaches  indefinitely  near  to  P,  then  the  limiting  position, 
TT,  of  the  secant  is  called  tlie  Tangent  to  the  curve  at  P. 


Fig.  25.  Fig.  26. 

The  tangent  TT'  is  said  to  touch  the  curve  at  P,  and  the 
point  P  is  called  the  Point  of  Contact. 

The  straight  line  PN  drawn  from  P,  perpendicular  to 
the  tangent  TT',  is  called  the  Normal  to  the  curve  at  P. 

Let  the  curve  be  referred  to  the  axes  OX,  OY,  and  let  M 
be  the  foot  of  the  ordinate  of  the  point  P.  Let  also  the 
tangent  and  the  normal  at  P  meet  the  axis  of  x  in  the 
points  T,  N,  respectively.  Then  TM  is  called  tlio  Sub- 
tangent  for  the  point  P,  and  MN  is  called  the  SubnormaL 

64.  To  find  the  equation  of  the  tangent  to  the  circle 
x^-\-y'^=^  r"^,  at  the  point  of  contact  (xi,  y^. 

Let  P  (Fig.  26)  be  the  point  (x^,  ?/i),  and  ()any  other  point 

(^2j  2/2)  of  the  circle.    Then  the  equation  of  the  secant  PQ  is 

y  —  lh^lh  —  yi 


0) 


78  ANALYTIC    GEOMETRY. 

Now,  since  (xi,  ij^)  and  (jv,  y.^  are  on  this  circle,  we  have 

Subtracting,        (x^^  —  oc^^)  +  (jh"  —  y^)  =  0. 
_  Factoring,  (a^  —  Xi)  (.r.,  +  x,)  +  (^,  —  y ,) (//,  +  ?/i)  =  0, 

Whence,  by  transposition  and  division,  we  have 

y^  —  yi  _  _  a-2  +  a'i 

a-2  —  a-i  v/2  +  y\ 

By  substituting  in  (1),  the  equation  of  the  secant  becomes 

y  ~  yi  _    •'^2  +  a^i 

Now  let  Q  coincide  with  P,  or  a-g^^u  y^^^Uu  ^^^^  secant 
becomes  a  tangent  at  P,  and  the  equation  becomes 

y  —  yi^     a-1 
^  —  ^1  Z/i' 

or  ^ia;  +  z/iZ/  =  •'^1^  +  2/1". 

And,  since  x^  -\-  yi^  =  r"^,  we  obtain 

Xi(ic+yii/  =  r%  [18] 

which  is  the  equation  required. 

NoTK.  If  we  had  put  Xn  =  xi,  1/2  =  2/1,  in  (1)  before  we  introduced 
the  condition  that  (xi,  2/1)  and  (X2,  2/2)  were  on  the  circle,  the  slope  of 

the  tangent  would  have  assumed  the  indeterminate  form  -• 

The  above  method  of  obtaining  the  equation  of  the  tangent  to  a 
circle  is  applicable  to  any  curve  whatever.  It  is  sometimes  called  the 
secant  method.  Equation  [18]  is  easily  remembered  from  its  sym- 
metry, and  because  it  may  be  formed  from  x^  +  y-  —  r^  by  merely 
changing  x-  to  XiX,  and  y"^  to  yiy. 


THE    CIRCLE.  79 

65.  To  find  tlie  P(p((tf'tmi  nf  the  normal  throxujh  (xj,  ?/i). 
The  sloi)e  of  the  tangent  is  — —  • 

Therefore,  that  of  the  normal  is  —  (§  45,  Cor.  2). 
Hence,  the  equation  of  the  normal  is  (§  38) 

which  reduces  to  the  form 

yix  -  XiU  =  O.  [19] 

Therefore,  the  normal  passes  through  the  centre. 

66.  To  find  the  equations  of  the  tangent  and  normal  to  the 
circle  (x  —  aY  -\-  (//  —  Ij)-  =  r  at  the  point  of  contact  (x^,  ij{). 

We  proceed  as  in  §  64,  only  now  the  equations  of  con- 
dition which  place  (.rj,  //j)  and  (.j-o,  ?/,)  on  the  circle  are 

(.ri-«)'+ 0/1 -*)'  =  '•". 

After  subtracting  and  factoring,  \\'e  have 

{x-x,)  {xA-x-2a)+{!,-y,)  (^,+^,-2i)=0, 

,  Vi  —  III  3*2  +  3"i  —  2a 

whence,  ^  = — —  • 

X2  —  X1  2/2  +  2/1— 2^> 

Hence,  the  equation  of  a  secant  tlirough  (x^.?/{)  and  (xn,  ?/.,)  is 

1/  —  2/1 .Tg  -{-Xi  —  2« 

.-»■  —  '^-1  y-i  +  //i  —  2^^ 

Making  .ro^.i,,  and  //2  =  //i,  and  reducing,  we  obtain 

(a?i  -  a)  {x-a)  +  {iji  -h){y-h)  =  r\  [20] 


80  ANALYTIC    GEOMETKY. 

Equation  [20]  may  be  immediately  formed  from  [18]  by 
affixing  —  a  to  the  x  factors  and  —  b  to  tlie  t/  factors,  on  the 
left-hand  side. 

By  proceeding  as  in  §  65,  we  obtain  for  the  equation  of 
the  normal 

{yi  -  b)  {X  -  aci)  -  (oci  -  a)  (y  -  yi)  =  O.  [21] 

67,  To  find  the  condition  that  the  straight  line  y  =  mx  +  c 
shall  touch  the  circle  x"-\-if=^  r^. 

I.  If  the  line  touches  the  circle,  it  is  evident  that  the  per- 
pendicular from  the  origin  to  the  line  must  be  equal  to  the 
radius  r  of  the  circle.     The  length  of  this  perpendicular  is 

— :^^z=:  (§  49).     Therefore,  the  required  condition  is 
Vl  -1"  nv^ 

c-  =  r^(l  +  «r). 

II.  By  eliminating  y  from  the  equations 

y  =  mx  -\-  c,     x^  +  ^^  =  ^, 
we  obtain  the  quadratic  in  x, 

(1  "1-  m^a;^  -|-  2mca;  =  v^  —  c^, 
the  two  roots  of  which  are 
mc 


Vy^(l  -f-  m^)  —  c' 
1  -\-  vi^  ~~  1  +  m^ 

If  these  roots  are  real  and  unequal,  the  line  will  cut  the 
circle;  if  they  are  equal,  it  will  touch  the  circle;  if  they  are 
imaginary,  it  will  not  meet  the  circle  at  all. 

The  roots  will  be  equal  if  Vr^(l  +  w^)  —  c^:=0;  that  is, 
if  c^=  r^(l -|-m^),  a  result  agreeing  with  that  previously 
obtained. 

If  in  the  equation  y^^mx-{-c  we  substitute  for  c  the 
value  7'Vl  -\-m^,  we  obtain  the  equation  of  the  tangent  of  a 
circle  in  the  useful  form 

y  =  mx  ±  »■  Vl  +  m^.  [22] 


THE    CIRCLE.  81 

This  equation,  if  we  regard  m  as  an  arbitrary  constant, 
represents  all  possible  tangents  to  the  circle  x'^-\-]f=^r^. 

Note  1.  Method  II  is  applicable  to  any  curve,  and  agi-ees  with  the 
definition  of  a  tangent  given  in  §  63. 

Note  2.  In  i^roblems  on  tangents  the  learner  should  consider 
whether  the  cooi'dinates  of  the  point  of  contact  are  involved.  If  they 
are,  he  should  use  equation  [18] ;  if  they  are  not,  then  in  general  it  is 
better  to  use  equation  [22]. 

Exercise  22. 

1.  Explain  the  meaning  of  the  double  sign  in  equation 
[22]. 

2.  Deduce  the  equations  of  the  tangent  and  normal  to 
the  circle  x'^-\-i/  =  t^,  assuming  that  the  normal  passes 
through  the  centre. 

3.  Find  the  equations  of  the  tangent  and  the  normal  to 
a;2-[-y2_.52  at  the  point  (4,  6).  Find,  also,  the  lengths  of 
the  tangent,  normal,  subtangent,  subnormal,  and  the  portion 
of  the  tangent  contained  between  the  axes. 

4.  A  straight  line  touches  the  circle  x^-\-if=r-  in  the 
point  (xi,  t/i).  Find  the  lengths  of  the  subtangent,  the  sub- 
normal, and  the  portion  of  the  line  contained  between  the 
axes. 

5.  What  is  the  equation  of  the  tangent  to  the  circle 
x^  -\-y^:=  250  at  the  point  whose  abscissa  is  9  and  ordinate 
negative  ? 

6.  Find  the  equations  of  tangents  to  x^-{-  ij^  =  10  at  the 
points  whose  common  abscissa  =  1. 

7.  Tangents  are  drawn  through  the  points  of  the  circle 
x^-{-y^  =  25  that  have  abscissas  numerically  equal  to  3. 
Prove  that  these  tangents  enclose  a  rhombus,  and  find  its  area. 

8.  The  subtangent  for  a  certain  point  of  a  circle  is  5^ ; 
the  subnormal  is  3.     What  is  the  equation  of  the  circle  ? 


82  ANALYTIC    GEOMETRY. 

Find  the  equation  of  the  straight  line  : 

9.    Touching  x~-\-;/-=232  at  the  point  whose  abscissa=14. 

10.  Touching  («  —  2)-  +  (//  —  3)'  =  10  at  the  point  (5, 4). 

11.  Touching  x^  -\-  if  —  3x  —  4?/  ^  0  at  the  origin. 

12.  Touching  x'^-^if  —  14x  —  4y/  —  5=0  at  the  point 
whose  abscissa  is  equal  to  10. 

What  is  the  equation  of  a  straight  line  toucliing  the 
circle  x^-\-y'^=  i^,  and  also  : 

13.  Passing  through  the  point  of  contact  (r,  0)? 

14.  Parallel  to  the  line  Ax-\-Bi/~\-  (7=0? 

15.  Perpendicular  to  the  line  Ax  -)-  /:>//+  C=  0  ? 

16.  Making  the  angle  45°  with  the  axis  of  x? 
.17.    Passing  through  tlie  exterior  point  (Ji,  0)  ? 

18.  Forming  with  the  axes  a  triangle  of  area  r^? 

19.  Find  the  equations  of  tlie  tangents  drawn  from  the 
point  (10,5)  to  the  circle  .t- -f" //"  =  100. 

20.  Find  the  equations  of  tangents  to  the  circle  x^-\-if 
+  10a;  —  Qnj  —  2  =  0  and  parallel  to  the  line  y  =  2x  —  7. 

21.  Find  the  lengths  of  the  subtangent  and  subnormal  in 
the  cii'cle  x^-\-if  —  l^x^  —  4// =  5  for  the  point  (10,9). 

22.  What  is  the  equation  of  the  circle  (centre  at  origin) 
that  is  touched  by  the  straight  line  x  cos  a-(- .V  sina:=/>? 
What  are  the  coordinates  of  the  point  of  contact? 

23.  When  Avill  tlie  line  Ax  -\-  B//—  C  =  0  touch  the  circle 
x^  +  ?/  =  v-^  ?    the  circle  (x  —  a.y  +  (y  —  f')'  ==  r^  ? 

24.  Find  the  equation  of  the  straight  line  toucliing  .t^+// 
=  ax-\-hy  and  passing  through  the  origin. 


THE    CIRCLE.  83 

Prove  that  the  following  circles  and  straight  lines  touch, 
and  tind  tlie  point  of  contact  in  each  case  : 

25.  x^ -{-  //'■^  -\-  ax  -\-b//^^0  and  ax  +  iy  +  «'■'  -\-b'^  =  0. 

26.  x^+i/-  —  2ax  —  2b y  -\-h^  =  {)  and  x  =  2a. 

27.  iK^H-  if  =  ax  -\-  bij  and  ax  —  bij  -\- 1!^  =  0. 

28.  What  is  the  equation  of  the  circle  (centre  at  origin) 
that  touches  the  Hue  i/  =  3x  —  5  ? 

29.  What  must  be  the  value  of  7)i  in  order  that  the  line 
t/  =  mx  -\- 10  may  touch  the  circle  x^-\-y^  =  100  ?  Show 
that  we  get  the  same  answer  for  the  line  y=^mx  — 10,  and 
explain  the  reason. 

y  30.  Determine  the  value  of  c  in  order  that  the  line 
ox  —  4//-l-c  =  0  may  touch  the  circle  a;^+^^  —  8a-  +  12y 
—  44-  =  0.     Explain  the  double  answer. 

31.  What  is  the  equation  of  the  circle  having  for  centre 
the  point  (5,  3)  and  touching  the  line  3x-\-2y  —  10  =  0  ? 

32.  What  is  the  equation  of  a  circle  whose  radius  =:  10, 
and  which  touches  the  line  4a;  -\-oy  —  70  =  0  in  the  point 
(10,  10)  ? 

33.  A  circle  touching  the  line  ■ix-\-oy-\-o=^0  in  the 
point  ( —  3,  3)  passes  through  the  point  (o,  9).  What  is 
its  equation  ? 

yT  34.    Under  what  condition  will  the  line  --f-^^l   touch 
the  circle  x^-\-y'^^^  r'  ? 

35.  What  is  the  equation  of  the  circle  inscribed  in  the 
triangle  whose  sides  are 

x  =  0,   y=0,    -  +  -^=1? 
a       0 


84  ANALYTIC    GEOMETRY. 

36.  Two  circles  touch  each  other  when  the  distance 
between  their  centres  is  equal  to  the  sum  or  the  difference 
of  their  radii.     Prove  that  the  circles 

touch  each  other,  and  find  the  equation  of  the  common 
tangent. 

37.  Two  circles  touch  each  other  when  the  length  of  their 
common  chord ^=0.    Find  the  length  of  the  common  chord  of 

(x-ay  +  (l/-by  =  r',     (x-by  +  (>/-ay  =  ^^, 

and  hence  prove  that  the  two  circles  touch  each  other  when 
(a  — by  =^2)^.  * 

Exercise  23.     (Review.) 
Find  the  radii  and  centres  of  the  following  circles  : 

1.  3x'  —  6x-\-3i/'-\-9i/  —  12  =  0. 

2.  7x^-  +  3>/  —  4u—(l  —  2xy^0. 
3-    y(l/  —  5)=x{3  —  x). 

4.  Vl  +  a'  (x'  +  /)  =  2b(x  +  uij). 

Find  the  equation  of  the  circle  : 

5.  Centre  (0,  0),  radius  =  9. 

6.  Centre  (7,  0),  radius  =3. 

7.  Centre  (—2,  5),  radius  =  10. 

8.  Centre  (3a,  4a),  radius  =  5a. 

9.  Centre  (b-\-c,  b  —  c),  radius  =  c. 

10.  Passing  through  {a,  0),  (0,  b),  (2a,  2b). 

11.  Passing  through  (0,  0),  (0,  12),  (5,  0). 

12.  Passing  through  (10,  9),  (4,  —5),  (0,  5). 

1 3.  Touching  each  axis  at  the  distance  — 7  from  the  origin. 


THE    CIRCLE.  85 

14.  Touching  both  axes,  and  radius  =r. 

15.  Centre  (a,  a),  and  cutting  chord  =  ^  from  each  axis. 

16.  Having  the  centre  (0,  0),  and  touching  ?/  =  2a;  -f-  3. 

17.  Having  the  centre  (1,  —  3),  and  touching  2x  —  y  =  4. 

18.  With  its  centre  in  the  line  5x  —  7y  —  8=0,  and 
touching  the  lines  2x  — 1/  =  0,  x  —  2y  —  6  =  0. 

19.  Passing  through  the  origin  and  the  points  common  to 
the  circles 

a^'  + /  -  6x  —  10?/  — 15  =  0, 
x''-\-u''-\-2x-\-   4?/-20  =  0. 

20.  Having  its  centre  in  the  line  ox  —  3)/  —  T  =  0,  and 
passing  through  the  points  common  to  the  same  circles  as 
in  No.  19. 

21.  Touching  the  axis  of  x,  and  passing  through  the 
points  common  to  the  circles 

x^-^7f -\- Ax- U/j- 68  =  0, 
x^  +  f  —  6x  —  22i/+30  =  0. 

22.  Find  the  centre  and  the  radius  of  the  circle  which 
passes  through  (9,  6),  (10,  5),  (3,  —2). 

23.  What  is  the  distance  from  the  centre  of  the  circle 
passing  through  (2,  0),  (8,  0),  (5,  9)  to  the  straight  line 
joining  (0,  — 11)  and  (— 16,  1)  ? 

24.  What  is  the  distance  from  the  centre  of  the  circle 
x^-\-y^  —  'ix-\-8>/=0to  the  line  ix  —  3y -f  30=  0  ? 

25.  What  portion  of  the  line  ?/ =  o./- +  2  is  contained 
within  the  circle  x^-\-y-  —  13a;  —  4y  —  9^0? 

26.  Through  that  point  of  the  circle  a;^  +  ^=25  for 
which  the  abscissa  =  4  and  the  ordinate  is  negative,  a 
straight  line  parallel  to  i/^ox  —  5  is  drawn.  Find  the 
length  of  the  intercepted  chord. 


86  ANALYTIC    GEOMETRY. 

27.  Through  the  point  (;/•,,  y^),  within  tlie  circle  x--\-y" 
=  r%  a  chord  is  drawn  so  as  to  l)o  bisected  at  this  \)oint. 
What  is  its  equation  ? 

28.  What  rehation  must  exist  among  the  coefficients  of 
the  equation  J  (x-  -\- ;/)  -\-  D.r  -\-  Ey  +  6'  =  0, 

(i)  in  order  that  the  circle  may  touch  the  axis  of  x  ? 
(ii)  in  order  that  the  circle  may  touch  thc^  axis  ot  y  ? 
(iii)  in  order  that  the  circle  may  touch  both  axes  ? 

29.  Under  what  condition  will  the  straight  line  ?/  =  wa; 
-{-  e  touch  the  circle  a?-  +  y-  =  2rx  ? 

30.  What  must  be  the  value  of  k  in  order  that  the  line 
3.x  +  4y  =  k  may  touch  the  circle  ?/  =  IOj;  —  x'-  ? 

31.  Find  the  equation  of  the  circle  that  passes  through 
the  origin  and  cuts  equal  lengths  a  from  the  lines  x=^y, 
x-\ry  =  0. 

32.  Find  the  equations  of  the  four  circles  whose  com- 
mon radius  =a\/'2,  and  which  cut  chords  from  each  axis 
equal  to  2a. 

33.  Fin'd  the  equation  of  the  circle  whose  diameter  is  the 
common  chord  of  the  circles  x"^ -{- y''^  =^  r"^,  (x. —  <i)--\-  y-=r-. 

Find  the  equation  of  the  straight  line  : 

34.  Passing  through  (0,  0)  and  the  centre  of  the  circle 

x'^-\-y'^=a(x-\-y). 

35.  Passing  through  the  centres  of  the  circles 

x'  +  y^  =  25  and  x"  +  //  +  C^x  —  8y  ==  0. 

36.  Passing  through  (0,  0)  and  touching  the  circle 

a^  -\-  y-  -  Crx  —  12//  +  41=0. 

37.  Parallel  to  a;  +  V3(^  — 12)  =  0  and  touching  x^  +  y- 
=  100. 


THE    CIRCLE.  87 

38.  Passing  through  the  points  common  to  the  circles 

X-  +  //-—    2x—    4y—    20  =  0, 
X-  +  /  —  14.r  —  IG//  +  100  =  0. 

39.  Prove  that  the  common  chord  of  the  circles  in  No.  38 
is  perpendicular  to  the  straight  Hue  joining  their  centres. 

40.  Find  the  area  of  the  triangle  formed  by  the  radii  of 
the  circle  cc^-j-  y^  =  169  drawn  to  the  points  whose  abscissas 
are  — 12  and  +  7  and  ordinates  positive,  and  the  chord  pass- 
ing through  the  same  two  points. 

41.  Prove  that  an  angle  inscribed  in  a  semicircle  is  a 
right  angle. 

42.  Prove  that  the  radius  of  a  circle  drawn  perpendicular 
to  a  chord  bisects  the  chord. 

43.  Find  the  inclination  to  the  axis  of  x  of  the  line  joining 
the  centres  of  the  circles  x^  -|-  2a:  +  y^  =  0,  x^  -\-  2i/  ■{- 1/~=0. 

44.  Determine  the  point  from  which  tangents  drawn  to 
the  circles       ^.2_|_y2_    2x  —    ()//+    G  =  0, 

a;'  +  f  —  22//  -  20a:  +  52  =  0, 
will  each  be  equal  to  4  VC. 

45.  Find  the  equations  of  the  circles  that  touch  the 
straight  lines  6.x  +  7//  +  9  =  0  and  7x  -\-6//-\-o^=0,  the 
latter  line  in  the  point  (3,  —  4). 

Obtain  and  discuss  the  equations  of  the  following  loci: 

46.  Locus  of  the  centre  of  a  circle  having  the  radius  r 
and  passing  through  the  point  (xi,i/i). 

47.  Locus  of  the  centre  of  a  circle  having  the  radius  r' 
and  touching  tlie  circle  (x  —  a)-  -\-  (//  —  1))"=  r. 

48.  Locus  of  all  points  from  which  tangents  drawn  to 
the  circle  (x  —  ('y-\-(!/  —  hy=^r^  have  a  given  length  t. 


88  ANALYTIC    GEOMETRY. 

49.  Locus  of  the  middle  point  of  a  chord  drawn  through 
a  fixed  point  ^  of  a  given  circle. 

50.  Locus  of  the  point  M  which  divides  the  chord  AC, 
drawn  through  the  fixed  point  J^  of  a  given  circle,  in  a 
given  ratio  AM :  MC  =  m  :  n. 

51.  Locus  of  a  point  whose  distances  from  two  fixed 
points,  A,  B,  are  in  a  constant  ratio  m  :  n. 

52.  Locus  of  a  point,  the  sum  of  the  squares  of  whose 
distances  from  two  fixed  points,  A,  B,  is  constant,  and 
equal  to  Jc\ 

53.  Locus  of  a  point,  the  difference  of  the  squares  of 
whose  distances  from  two  fixed  points,  A,  B,  is  constant 
and  equal  to  k^. 

54.  Locus  of  the  middle  point  of  a  line  of  constant 
length  d  which  moves  so  that  its  ends  always  touch  two 
fixed  perpendicular  lines. 

55.  Locus  of  the  vertex  of  a  triangle  whose  base  is  fixed 
and  of  constant  length,  and  the  angle  at  the  vertex  is  also 
constant. 

56.  One  side,  AB,  of  a  triangle  is  constant  in  length  and 
fixed  in  position;  another  side,  AC,  is  constant  in  length 
but  revolves  about  the  point  A.  Find  the  locus  of  the 
middle  point  of  the  third  side,  BC. 

57.  Find  the  locus  of  the  intersections  of  tangents  at  the 
extremities  of  a  chord  whose  length  is  constant. 

58.  A  and  B  are  two  fixed  points,  and  the  point  P  moves 
so  that  PA  =  n  X  PB ;  find  the  locus  of  P. 


THE    CIRCLE. 


89 


SUPPLEMENTARY    PROPOSITIONS. 

68.  A  Diameter  of  a  curve  is  the  locus  of  the  middle 
points  of  a  system  of  parallel  chords.  The  chords  which 
any  diameter  bisects  are  called  the  Chords  of  that  diameter. 

69.  To  find   the   equation  of  a  diameter  of  the   circle 


Fig.  27. 


Let  the  equation  of  any  one  of  the  parallel  chords  (Fig. 
27)  he  7/  =  mx  +  c,  and  let  it  meet  the  circle  in  the  points 
(xi,i/i)  a,nd(x2,1/o). 

Then  (§§  37  and  64)  m  =  -  ^^^^-  (1) 


2/1  +  2/2 

Let  (x,i/)  be  the  middle  point  of  the  chord;  then  2x^= 
Xi-\-x2,  2_y  =  ?/i -f  2/2  (§  8),  and  by  substitution  we  have 


or 


x 

vi  = , 

y 
1 

V = — ^) 


(2) 


a  relation  which  evidently  holds  true  for  the  middle  points 
of  all  the  chords.  Therefore  (2)  is  the  equation  of  a  diameter. 
Cor.  From  (2)  we  see  that  a  diameter  of  a  circle  is  a 
straight  line  passing  through  the  centre  and  perpendicular 
to  its  chords. 


90 


ANALYTIC    GEOMKTRY. 


70.  Two  distinct,  tiro  coiiicidcitt,  or  no  tangents  can  he 
i/raivn  to  a  circln  tlinnKjIi  anij  point  (A,  k),  according  us  this 
jjoint  is  ivithout,  on,  uf  ivlthin  the  circle. 

Let  the  tancrent 


(J  =  inx  -f-  /•  Vl  +  ni^ 
j)uss  through  the  point  (/;,  /.);  then 

mli  -\-r  '\ll  -\-  ni\ 


k 


Transposing  and  squaring,  we  have 

{U'—r'')ni''  —  2hkm=i^- 


k\ 


hk  ±  r  \J li' +  Ic' —  r'' 


h' 


(1) 


The  values  of  m  given  in  (1)  are  the  slopes  of  the  tan- 
gents through  (Ji,k).  Now,  these  values  are  real  and 
unequal,  real  and  equal,  or  imaginary,  according  as  ]i^-\-k^ 
>,  =,  or  <C  r^;  that  is,  according  as  (A,  k)  is  without,  on, 
or  within  the  circle.  Hence,  two  distinct,  two  coincident, 
or  no  tangents  can  be  drawn  througli  (/<,  k),  according  as 
(Ji ,  k)  is  without,  on,  or  within  the  circle. 


Fig.  28. 

71.    To  find  the  equation  of  the  chord  joining  the  points  of 
contact  of  the  two  tangents  from  any  external  jjoint  (Ji,  Jc). 


THE    CIRCLE.  '  91 

Let  (rTj,  ?/i),  (.ro,  7/2)  I'f'  tlu;  points  of  contact  (^  and  7i'  ; 
then  the  equations  ol'  tlie  tangents  FQ  and  PR  are  (§  G4) 

Since  both   tangents  pass   tlirough  ./'(A,  h),  botli  these 

equations  are  satisfied  by  the  coordinates  h,  k  ;  therefore, 

hxi  +  hji  =  r,  (1) 

hx^  +  hj^  =  r\  (2) 

From  equations  (1)  and  (2)  we  see  that  tlie  coordinates  of 

both  the  points  (./'i,  y^),  (x^,  1/2)  satisfy  the  equation 

hx-\-kt/  =  7^.  (3) 

Hence,  the  locus  of  (3),  wliich  is  a  right  line,  passes  through 

both  points  of  contact  ;  and,  therefore,  (3)  is  the  eqnation  of 

the  chord  QR.    The  chord  QR  is  called  the  Chord  of  Contact. 

72.  Sii.pjiose  a  eliord  of  a  cirrlc.  to  turn  round  an//  fixed 
point  (h,  k)  ;  to  find  the  lorvs  of  tlie  intei'section  of  the  two 
tangents  drawn  at  its  extremities. 

Let  P  (Fig.  29  or  30)  be  the  fixed  point  (A,  k),  QPR  one 
position  of  the  revolving  chord,  and  let  the  tangents  at  Q 
and  R  intei-sect  in  Pj,  (.rj,  7/,)  ;  it  is  required  tb  find  the 
locus  of  Pi  as  the  chord  turns  about  P.  Since  QR  is  the 
chord  of  contact  of  tangents  drawn  from  the  point  P, 
(•*^i)  2/i))  its  equation  is  (§  71) 

3-1^  + 7/1// ='•''.  (1) 

Since  (1)  passes  through  (//,  /,•),  we  have 

hx,  +  k;,,  =  r\  (2) 

But  (x^,  7/j)  is  a7i7/  point  in  tlio  required  locus,  and  by  (2) 
its  coordinates  satisfy  the  eqnation 

'   />,--{- ki/=r"-;  (3) 

hence,  (3)  is  the  eqnation  of  tlie  recpiirc^d  locus. 

Since  (3)  is  of  the  first  degree,  the  locus  is  a  straight  line. 


92 


ANALYTIC    GEOMETKT. 


The  line  hx  +  kij  =  r-  is  called  the  Polar  of  the  point  (h,  k) 
with  regard  to  the  circle  x^-{-7/=  7^,  and  the  point  (h,  k)  is 
called  the  Pole  of  the  line.  The  pole  (A,  k)  may  be  with- 
out, on,  or  within  the  curve.  In  Fig.  29  it  is  within,  while 
in  Fig.  30  it  is  without  the  circle. 


Fig.  29. 


THE    CIRCLE.  93 

Cor.  1.  If  the  point  (h,  k)  is  on  the  circle,  (3)  is  evi- 
dently the  equation  of  the  tangent  at  (A,  k) ;  hence, 

The  polar  of  any  point  on  the  circle  is  identical  with  the 
tangent  at  that  2^0  int. 

Cor.  2.  If  (Ji,  k)  is  an  external  point,  by  §  71,  (3)  is  the 
equation  of  the  chord  of  contact  of  tangents  from  (h,  k)  to 
the  circle  ;  hence, 

The  polar  of  any  external  point  is  the  same  line  as  the 
chord  of  contact  of  tangents  draivn  from  that  point. 

Thus,  in  Fig.  30,  iOf  is  the  polar  of  P,  or  the  chord  of 
contact  of  tangents  drawn  from  P. 

73.  The  polar  and  pole  of  a  circle  may  be  defined  as 
follows  :  If  a  chord  of  a  circle  is  turned  round  a  fixed 
point  (Ji,  k),  the  locus  of  the  intersection  of  the  two  tan- 
gents at  its  extremities  is  the  polar  of  the  pole  (h,  k)  with 
regard  to  that  curve. 

74.  If  the  polar  of  a  point  F  passes  through  P',  then  the 
polar  of  P'  loill  pass  through  P. 

Let  P  be  the  point  (A,  k~),  P'  the  point  (A',  k'),  and  let  the 
equation  of  the  circle  be  x-  -{-  y'^=^  1^. 

Then  the  equations  of  the  polars  of  P  and  P'  are 

hx-\-ky=i^  (1) 

h'x  +  k'y  =  i^.  (2) 

If  P'  is  on  the  polar  of  P,  its  coordinates  must  satisfy 
equation  (1)  ;  therefore, 

hh'-\-kk'  =  i^. 

But  this  is  also  the  condition  that  P  shall  be  on  the  line 
represented  by  (2)  ;  that  is,  on  the  polar  of  P'.  Therefore, 
P  is  on  the  polar  of  P'. 

This  relation  of  poles  and  polars  is  illustrated  in  the 
Figs.  29  and  30. 


94 


ANALYTIC    GEOMKTUY. 


75.    To  find  a  (jeometvical  conslvnction  for  the  polar  of  a 
point  with  respect  to  a  circle. 


Fig.  32. 

The  equation  of  the  line  through  any  point  P  (Ji,  k)  and 
the  centre  of  the  circle,  or  the  origin,  is 

hx—hi/  =  0.  (1) 

Now,  the  equation  of  tlie  polar  of  P  is 

/'.«  +  %=>^.  (2) 

But  the  loci  of  (l)and  (U)  are  perpendicular  (§  45,  Cor.  2). 
Hence,  if  PC  is  the  polar  of  P,  OP  is  perpendicular  to  BC, 
and 


00  = 


^h^+Jc" 


(§41) 


Also,  OP^-sJh^'+k-. 

Therefore,  OPXOQ  =  A 

Hence,  to  construct  the  polar  of  P  : 

Join   OP,  and  let  it  cut  the  circle  in  A  ;  take  Q  in  the 
line  OP,  so  that 

OP  :  0A  =  OA  :  OQ. 

Tlie  line  through  Q  per^x'ndicular  to  OP  is  tlie  ]iolar  of  P. 

To  locate  the  pole  of  BC,  draw  OQ  perpendicular  to  BC, 
andtakePsoth.t        oQ:OA=OA:OP. 


TIIK    CIRCLE. 


95 


76.  To  find  the  length  of  tJie  tangent  drawn  from  any 
point  (h,  k)  to  the  circle     (x  —  ay-\-(g  —  Of — >'^'  =  0,       (1) 

Let  F  (Fig.  33)  be  the  point  (h,  k),  Q  the  [)oiiit  of  contact, 
C  the  centre  of  the  circle;  then,  since  FQC  is  a  riglit  angle, 

Fq'=fc^—qc\ 

Now,  FC'  =  (h  -  a)-  +  (/.■  -  by,  and  'QC'  =  i\ 

Therefore,  FQ-  =  (h  —  a)"-  +  (k  -  hf  —  i^. 

Hence  FQ^  is  fonnd  by  simply  substituting  the  coordinates 
of  F  for  X  and  xj  in  the  expression  {x  —  0)''^+  {ij  —  hf  —  v"^. 

P 


Fig.  33. 


Fig.  34. 


If  for  brevity  we  write  <S'  instead  of  (.r  —  af  -\-  {ji  —  hy  —  r', 
then  the  equation  ;iS  =  0  will  represent  the  general  equation 
of  the  circle  after  division  by  the  common  coefficient  of  x^ 
and  ?/,  and  we  may  state  the  above  result  as  follows : 

If  )S^0  is  the  erp/atioji  of  a  circle,  and  the  coordinates  of 
any  point  are  suhstitnted  for  x  and  y  in  S,  the  result  will  he 
equal  to  the  sqnare  of  the  length  of  the  tangent  drawn  from 
the  point  to  the  circle. 

11.  To  find  the  locus  of  the  point  from  n'hich  tangents 
drawn  to  tu'o  given  circles  are  equal. 

Let  the  equations  of  tlic  circles  O  and  O'  (Fig.  34),  be 
(.■-«)^+0/-A)'-'-'=0,  (1) 

and  (x  —  a'y  -\-(y  —  l>'y  —  r'^=0.  (2) 


96  ANALYTIC    GEOMETRY. 

Then,  if  the  tangents  drawn  from  P  (x,  y)  to  the  circles 
(1)  and  (2)  are  eq^ual,  we  have 

{x  —  ay-\-{y-hy-  r2  ={x  —  a'f  -{-{y-b'f-  r'%  (3) 
which  is  the  equation  of  the  required  locus. 

Cor.  1.  Performing  the  indicated  operations  in  (3),  and 
transposing,  we  have 

2  («  —  a')  ic  +  2  {b  —  b')  y  =  a'  —  a'^^b^  —  b'^  —  ',-^r''',    (4) 
which  shows  that  the  locus  is  a  straight  line. 

This  locus  is  called  the  Radical  Axis  of  the  two  circles. 

Hence,  if  S-i  =  0,  82,=^  0  are  the  equations  of  two  circles,  then 

Si  =  025  or  Si  —  *S'2  =^^  0, 
7vill  be  the  equation  of  their  radical  axis. 

Cor.  2.  When  the  circles  *S'i  =  0  and  *S'2=:0  intersect,  the 
locus  of  Si  =  S2  passes  through  their  common  points. 

Hence,  ivhen  two  circles  intersect  or  are  tangent,  their 
radical  axis  is  their  common  chord  or  common  tangent. 

CoR.  3.  The  slope  of  (4)  is  the  negative  reciprocal  of  the 
slope  of  the  line  joining  the  centres  of  (1)  and  (2). 

Hence,  the  radical  axis  of  two  circles  is  ji&i'petidicular  to 
the  line  joining  their  centres. 

78.    Let  >S'=0,  *S'i  =  0,  *S^2  =  0  be  the  equations  of  three 
circles,  in  each  of  which  the  coefficient  of  x^  is  unity. 
Then  the  equations  of  their  radical  axes,  taken  in  pairs,  are 

S  —  Si  =  U,      Si  —  02  ■""  ^t      s  —  S2  ^=  0. 

The  values  of  x  and  y  that  will  satisfy  any  two  of  these 
equations  will  also  satisfy  the  third.  Therefore,  the  third  axis 
passes  through  the  point  common  to  the  other  two.    Hence, 

The  three  radical  axes  of  three  circles,  taken  in  pairs,  vieet 
in  a  point.  This  point  is  called  the  Radical  Centre  of  the 
three  circles. 


THE    CIRCLE,  97 


Exercise  24. 


1.  What  is  the  equation  of  the  diameter  of  the  circle 
x^  -{-  7j-  =  20    that    bisects    chords    parallel    to    the    line 

2.  What  is  the  equation  of  the  diameter  of  the  circle  that 
bisects  all  chords  whose  inclination  to  the  axis  of  x  is  135°  ? 

3.  Prove  that  the  tangents  at  the  extremities  of  a 
diameter  are  parallel. 

4.  Write  the  equations  of  the  chords  of  contact  in  the 
circle  x^ -\- y^  =  7'^  fov  tangents  drawn  from  the  following 
points:  (r,  r),  {2r,  3?'),  {a-\-h,  a  —  b). 

5.  From  the  point  (13,  2)  tangents  are  drawn  to  the  circle 
cc-  +  y^  =  49  ;  what  is  the  equation  of  the  chord  of  contact  ? 

6.  What  line  is  represented  by  the  equation  hx  +  ky=^  ?'^ 
when  (Ji,  k)  is  on  the  circle  ? 

7.  Write  the  equations  of  the  polars  of  the  following 
points  with  respect  to  the  circle  a:^  -f  ?/-  ^  4 : 

(i)(2,3).         (ii)(3,-l).         (iii)(l,-l). 

8.  Find  the  poles  of  the  following  lines  with  respect  to 
the  circle  x^-{-y^=^?>5: 

(i)  4a;  +  6?/  ==  7.  (ii)  3a;  —  2y  =  o.  (iii)  ax  +  h)j  =  1 . 

9.  Find  the  pole  of  3x-\-4:y  =  7  with  respect  to  the 
circle  x--\-  y-=  14. 

10.  Find  the  pole  of  Ax-\-By-\-C^O  with  respect  to 
the  circle  x^-\-  y-  =  r. 

11.  Find  the  coordinates  of  the  points  in  which  the  line 
a;  =  4  cuts  the  circle  x^  +  y  =  2o;  also  find  the  equations 
of  the  tangents  at  those  points,  and  show  that  they  inter- 
sect in  the  point  i^^-,  0). 


98  anai.vtk;   gkomktrv. 

12.  If  tlio  polavs  of  two  points  /',  Q  meet  in  B,  then  R 
is  the  pole  of  the  line  FQ. 

13.  If  the  polar  of  (A,  k)  witli  respect  to  tlie  circle 
3.2_j_y2_.j.2  touches  the  circle  x^-\->f  =  2rx,  then  k'^-\-'2rh 

14.  If  the  polar  of  (A,  k)  with  respect  to  the  circle 
a:^ -\- f/^  =  c^  touches  the  circle  4(^x^-\-y^)^c'^,  then  the  pole 
(h,  k)  will  lie  on  the  circle  a;^+?/-  =  4cl 

15.  Find  the  polar  of  the  centre  of  tlie  circle  x'^  -\-  //-  =  r^. 
Trace  the  changes  in  the  position  of  the  polar  as  tlie  pole  is 
supposed  to  move  from  the  centre  to  an  infinite  distance. 

16.  Wliat  is  the  square  of  the  length  of  the  tangent 
drawn  from  tlie  point  (Ji,  k)  to  the  circle  .t"  +  //' ^  ?•- ? 

17.  Find  the  length  of  the  tangent  drawn  from  (2,  5)  to 
the  circle  x^-  +  f  —  2x  — ,%  —  1  =  0. 

Find  the  radical  axis  of  the  circles : 

18.  (.r  +  5)^  +  (y  +  6)~o,  (x-7y-\-(>/-ny-=i6. 

1 9.  .7-2  +  ff  -f  2x  +  3//  —  7  =  0,  X-  +  //•-'  —  2x  —  7/4-1=0. 

20.  X-  +  f  +  hx  +  />//  —  c  =  0,  ax"^  +  air  +  ""->  +  />'//  =  ^^■ 

21.  Find  the  radical  axis  and  length  of  tlie  common 
choiwl  of  tlie  circles 

.r,"  4-  y'  -\-  (IX  -\-  Jnj  -|-  *"  =  0,  :i^  -\-  //-  -|-  hx  -f-  (nj  4-  c  =  0. 

22.  Find  the  radical  centre  of  the  three  circles 

a;2  +  yH  4x  +  7  =  0, 

2.^2+ LY-  +  o.r  +  5/y  + 9  =  0, 

a;^-|- ?/"-(- //  =  0. 


CHAPTER  IV. 

DIFFERENT   SYSTEMS   OF   COORDINATES. 

Rectilinear  System. 

79.  When  we  define  the  position  of  a  point,  with  refer- 
ence to  any  fixed  lines  or  points,  we  are  said  to  nse  a 
System  of  Coordinates. 

In  the  Rectilinear  System,  already  described,  we  liave 
thus  far  employed  only  rectangular  axes,  or  coordinates, 
which  are  to  be  preferred  for  most  purposes,  on  account 
of  their  greater  simplicity.  When  the  axes  of  reference 
intersect  at  oblique  angles,  the  axes  and  coordinates  are 
called  Oblique. 

,Y 


Let  OX,  OF  (Fig.  35)  be  two  axes  making  an  acute  angle, 
A'OF=w,  with  each  other.  If  we  draw  PN^  to  OX,  and 
PJ/||  to  OY,  then  the  coordinates  of  P  are 

NP  =  0M=  X,     MP  =  //. 

Since  oblique  and  rectangular  coordinates  differ  only  in 
the  angle  included  between  the  axes,  any  of  the  previously 
deduced  formulas  that  do  not  depend  on  any  property  of 
the  right  angle  are  apjdicable  wlien  the  axes  are  oblicjue. 
Thus,  formulas  [2],  [3],  [4],  [7J  liold  for  oblique  axes  as 


100  ANALYTIC    GEOMETRY. 

well  as  for  rectangular,  and  therefore  are  general  formulas 
for  the  Rectilinear  System. 

When  the  axes  are  oblique,  instead  of  [1],  we  evidently 
have  (Fig.  35) 

PQ  =  \IpW  +  UQ'  —  2PR  XBQ  cos  PBQ, 

•••  d=  '^ (^2  —  xiY  +  (yz  —  yif  -\-  ''^ix2  —  ^i)  (yz  — yi)cosa>, 

which  reduces  to  [1]  when  w=:90°. 

The  Rectilinear  System  is  sometimes  called  the  Cartesian 
System,  from  Descartes,  who  first  used  it. 

80.  To  find  the  equation  of  the  straight  line  AC,  referred 
to  the  oblique  axes  OX,  OF  (Fig.  36),  having  given  the  inter- 
cept OB=^b  and  the  angle  XAC=y. 

Let  P  be  any  point  (x,  y)  of  the  line.  Draw  BD  ||  to 
OX,  meeting  PM  in  D.     Then,  by  Trigonometry, 


PD sin  y  y  —  h sin  y 

BD      sin  (a»  —  y)'  X         sin  (w  —  y) 

If  now  we  put  m  =  —. — ; — - — r ,  we  obtain  as  the  result  an 
sin  ((!>  —  y) 

equation  of  the  same  form  as  [6],  p.  38, 

y  =  mx  -\-  b. 

Here  771  =  the  ratio  of  the  sines  of  the  angles  which  the 

line  AC  makes  with  the  axes  ;  that  is,  m  =  sin  JT^P-^  sin 

PBY,  which  equals  tan  X^P^tan  y  when  w  =  90°. 


DIFFERENT  SYSTEMS  OF  COORDINATES. 


101 


81.  Oblique  coordinates  are  seldom  used,  because  they 
generally  lead  to  more  complex  formulas  than  rectangular 
coordinates.  In  many  cases,  however,  they  may  be  em- 
ployed to  advantage.  An  example  of  this  kind  is  furnished 
by  problem  No.  23,  p.  65 : 

To  prove  that  the  medians  of  a  triangle  meet  in  one  point. 

If  a,  b,  c  represent  the  three  sides  of  the  triangle,  and  we 
take  as  axes  the  sides  a  and  b,  then  the  equations  of  the 
sides  and  also  of  the  medians  may  be  written  with  great 
ease,  as  follows : 

The  sides,       !/  =  0,    x  =  0,   -  +  {^  =  1. 


^0,  '-+■''■■ 

a      b 


The  medians, 


a        0  a        b 


1=0, 


a      b 


On  comparing  the  equations  of  the  medians,  we  see  that 
if  we  subtract  the  second  equation  from  the  first,  we  obtain 
the  third;  therefore,  the  three  medians  must  pass  through 
the  same  point  (§  53). 

Polar  System  of  Coordinates. 
82.    Next  to  the  rectilinear,  tlie  system  of  coordinates 
most  frequently  used  is  the  Polar  System. 


Fig.  37 


Let  0  (Fig.  37)  be  a  fixed  point,  OA  a  fixed  straight  line, 
P  any  point.     Join  OP. 


102  ANALYTIC    GEOMETRY. 

It  is  evident  tliat  we  know  the  position  of  P,  provided 
we  know  the  distance  OP  and  the  angle  wliieh  OT  I'oi'uis 
with  OA. 

Thus,  if  we  denote  the  distance  OP  by  p,  and  tlie  angle 
AOP  by  6,  the  position  of  P  is  determined  if  p  and  6  are 
known. 

The  fixed  point  O  is  called  the  Pole,  an<l  the  fixed  line 
OA  the  Polar  Axis;  p  and  6  are  called  the  Polar  Coordinates 
of  P;  p,  its  Radius  Vector;  and^,  its  Direction  or  Vectorial 
Angle. 


Every  point  in  a  plane  is  perfectly  determined  by  a  posi- 
tive value  of  p  between  0  and  oo,  and  a  positive  value  of  6 
between  0°  and  3G0°  (or  0  and  27r,  circular  measure).  P>ut, 
in  order  to  represent  by  a  single  equation  all  the  points  of 
a  geometric  locus,  we  often  employ  negative  valnes  of  p  and 
6,  and  adopt  the  following  laws  of  signs  : 

(i)  6  is  positive  when  measured  from  right  to  left,  and 
negative  when  measured  in  tlie  opposite  direction. 

(ii)  p  is  positive  or  negative  according  as  it  extends  in 
the  direction  of  the  terminal  side  of  6  or  in  the  opposite 
direction.  Thus,  any  given  point  may  be  determined  in 
four  different  ways. 


DIFFERENT    SYSTEMS    OF    COOKDINATES. 


103 


Fur  example,  suppose  tluit  the  straight  line  r()l\  bisects 
the  first  and  third  quadrants,  and  that  in  this  line  we  take 
points  F,  Pi,  at  the  same  distance  OF  =  a  from  O ;  then 

F  is  the  point(//,  i.7r)or( — a,|7r)or(— «, —  ^7r)or(     a, —  In); 
Fi  is  the  point  (^A,  ^Tr)o\-{—a,\Tr)  or(    a,  — ^7r)()r(— «,  — ;j7r). 


Fig.  39. 


Fig.  40. 


'Y    83.     To  Jiml  the  jJolar  equation  of  a  circle. 

(i)  Let  the  pole  0  be  at  the  centre  (Fig.  ,38).  Then,  if 
r  denotes  the  radius,  the  polar  equation  is  simply  p=^r. 

(ii)  Let  the  pole  0  be  on  the  circumference  (Fig.  o*.)), 
and  let  the  diameter  OF  make  an  angle  a  with  the  initial 
line  OA.    Let  F  be  any  point  (p,  6)  of  the  circle.     Join  BF. 

Then,  OF=^OD  cos  EOF, 

or  P  =  2»'  cos  (9  -  a).  [23] 

If  OB  is  taken  as  the  initial  line,  the  equation  becomes 

P  =  2r  cos  9.  [24] 

(iii)  Let  the  pole  0  be  any  point,  and  the  centre  the 
point  (p',  6').     Then  in  the  triangle  OCF  (Fig.  40), 

0F~  —  20FX  OCXcos  COF-]- OC^— aF-  =  Q, 

or  9-  -  2pp'  cos  (8  -  6')  +  p'-2  -  r'^  =  O,  [2r)] 

the  most  general  form  of  tlie  polar  equation  of  a  circle. 


104  ANALYTIC    GEOMETRY. 

Exercise  25. 

1 .  Find  the  distances  from  the  point  P  in  Fig.  38  to  the 
two  axes. 

2.  Prove  that  the  equation  of  a  straight  line,  referred  to 
oblique  axes  in  terms  of  its  intercepts,  is  identical  in  form 
with  [7],  p.  39. 

3.  If  the  straight  line  P2O-P3  (Fig.  38)  bisects  the  second 
and  fourth  quadrants,  what  are  the  polar  coordinates  of  the 
points  P2  and  P3  ?  Give  more  than  one  set  of  values  in 
each  case. 

4.  Construct  the  following  points  (on  paper,  take  a  ^  1  in.) : 

«,  0  V   [a,  IT  ),   [<i,— 


9 
2„,|),  (2„,.),  (^„cos|| ),  (.,|),  (^S»,| 

—  3rt,  —  I,   (  4rt,  tan~^ -  ),  I  4a,  tan-^ - 

'  3/   V  3/  V  4 

4 
Note.     The  expression  tan— 1  -  in  higher  Mathematics  means  the 

o 

4 

angle  whose  tangent  is  -■ 

5.  If  pi,  p2  denote  the  two  values  of  p  in  equation  [25], 
p.  103,  prove  that  pip2  =  p'^  —  '■^-  What  theorem  of  Ele- 
mentary Geometry  is  expressed  by  this  equation  if  the  pole 
is  outside  the  circle  ?  if  the  pole  is  inside  the  circle  ? 

6.  Through  a  fixed  point  P  in  a  circle  a  chord  PP  is 
drawn,  and  then  revolved  about  P;  find  the  locus  of  its 
middle  point. 

Note.  In  such  problems  as  this  there  is  a  great  advantage  in 
using  polar  equations. 

7.  li p  denotes  the  distance  from  the  pole  to  a  straight 
line,  a  the  angle  between  p  and  the  polar  axis,  prove  that 
the  polar  equation  of  the  line  is  p  cos  (0  —  a)  =p. 


DIFFERENT    SYSTEMS    OF    COORDINATES. 


105 


Transformation  of  Coordinates. 

84.  The  equation  of  a  curve  is  oftentimes  greatly  sim- 
plified by  referring  it  to  a  new  set  of  axes,  or  to  a  new 
system  of  coordinates.  For  example,  compare  equations 
[15]  and  [16],  p.  71.  Hence,  it  is  sometimes  useful  to 
be  able  to  deduce  from  the  equation  of  a  curve  referred 
to  one  set  of  axes  or  to  one  system  of  coordinates,  its 
equation  when  referred  to  another  set  of  axes  or  to  an- 
other system  of  coordinates.  Either  of  these  operations 
is  known  as  the  Transformation  of  Coordinates.  It  consists 
of  expressing  the  old  coordinates  in  terms  of  the  new,  and 
then  replacing  in  the  equation  of  the  curve  the  old  coordi- 
nates by  their  values  in  terms  of  the  new ;  we  thus  obtain 
a  constant  relation  between  the  new  coordinates,  that  will 
represent  the  curve  referred  to  the  new  axes  or  system. 

85.  To  change  the,  origin  to  the  •point  (Ji,  k)  without 
changing  the  direction  of  the  axes. 


Y 

Y' 

f 

w 

0 

r 

0 

- 

i 

M 

X 

Fig.  41. 

Let  OX,  OF  be  the  old  axes,  O'X',  O'Y'  the  new;  and 
let  (x,  y),  (x',  y')  be  the  coordinates  of  the  same  point  P, 
referred  to  the  old  and  new  axes  respectively. 

Then  (Fig.  41) 
OA  =  h,  AO'^k,   OM=x,   MP^y,   0'j\r  =  x\  MP  =  y'. 
x=OA    +  AM^  0A-{-  O'M'  =  x' -\-  h. 
y  =  MM'  +  MP  =AO'-Y  MP  =  u'-{-k. 


106 


ANALYTIC    GEOMETRY. 


These  relations  are  equally  true  ior  rectangular  and 
oblique  coordinates. 

Hence,  to  find  what  the  equation  of  a  curve  becomes 
wlieu  the  origin  is  transferred  to  a  point  (Ji,  k),  the  new 
axes  running  parallel  to  the  old,  we  must  substitute  for  x 
and  1/  the  values  given  above. 

After  the  substitution,  we  may  write  x  and  //  instead  of 
x'  and  y' ;  so  that  })ractically  the  change  is  eifected  by 
siinjjli/  ivrltiny  x-\-  h  in  jjlt^f-ce  of  x,  y-\-h  i/ij^lcice  of  y. 

If,  however,  we  wish  to  transform  a  ^>t»m^  (x,  y)  from 
the  new  to  the  old  system,  we  must  write  x  —  It  in  place 
of  x  and  y  —  k  in  place  of  y. 

86.  To  chanye  the  reference  of  a  curve  from  one  set  of 
rectanyidar  axes  to  another,  the  orhjin  remainintj  the  same. 


Let  (a:,  y)  be  a  point  P  referred  to  the  old  axes  OX,  0  Y; 
(x',  y')  the  same  point  referred  to  the  new  axes  OA"',  OY 
(Fig.  42).     Then 

03f=x,  MP  =  y,  ON=x',  Nr  =  y'. 

Let  the  angle  XOX'  ==  6.  Draw  NQ,  NR  i.  to  PM,  OX, 
respectively;  tlien 

NPQ=  QXO  =  EON=  e. 


DIKKKIJENT    SYSTEMS    OF    COOliDIN A'I'KS.  107 

Hence,  0M=  OR  —  RM=  OR  —  NQ  =  ON  cos  6  —  FN  si  ii  6. 
Or  x^=x'  COS  6  —  y'  sin  6. 

And     PJ/=  MQ  +  QF  =  KN-\-  QF  =  ON  sin  Q  +  7'.V  cos  ^. 
Or  y^=x'  sin  ^  +  y'  cos  6. 

Therefore,  to  find  what  the  equation  of  a  curve  becomes 
when  referred  to  the  new  axes,  we  must  write 

X  cos  6  —  1/  sin  6  for  x,     x  sin.  6  -\-  //  cos  6  for  y. 

"TT"^  87.  To  change  the  reference  of  a  curve  from  one  set  of 
rectangular  axes  to  another,  both  the  origin  and  the  direction 
of  the  axes  being  changed. 

First  transform  the  equation  to  axes  through  the  new 
origin,  parallel  to  the  old  axes.  Then  turn  these  axes 
through  the  required  angle. 

If  (Ji,  k)  is  the  new  origin  referred  to  the  old  axes,  0  the 
angle  between  the  old  and  new  axes  of  x,  we  obtain  as  the 
values  of  x  and  y  for  any  point  P,  in  terms  of  the  new 
coordinates, 

a*  =  A  4"  x'  cos  6  —  y'  sin  6, 
y^k-{-x'  sin  6 -\- y'  cos  6. 

In  making  all  these  transformations,  attention  must  be 
paid  to  the  signs  of  h,  k,  and  6. 

88.  To  change  the  reference  of  a  curve  from,  rectangular 
to  oblique  axes,  the  origin  remaining  the  same. 

Let  a,  ^  be  the  angles  formed  bj''  the  2iositive  directions 
of  the  new  axes  OX',  OY'  (Fig.  43)  with  the  positive 
direction  of  OX  Let  tlie  old  coordinates  of  a  point  P 
be  x,  y;  and  the  new  coordinates,  x\  //'.  Then  from  the 
right  triangles  ORN,  PQN  we  readily  obtain  tlie  formulas 
X  =  x'  cos  a  +  y'  cos  (3, 
y  =  x'  sin  a-\-y'  sin  (3. 

Investigate  the  special  case  when  fS=^a-\-[)0°. 


108 


ANALYTIC    GEOMKTltV. 


89.  To  deduce  the  formulas  for  finding  the  polar  equation 
of  a  curoe  from  its  rectangular  equation. 

Let  cc,  y  be  the  rectangular  coordinates  of  any  point  P, 
and  p,  6  its  polar  coordinates. 


Y 

'ix-^ 

\ 

vr 

^^ 

0 

M 

R    X 

Y 

/P 

/      ^^-^"^^^^ 

0 

M   X 

Fig.  43. 


Fig.  44. 


(i)  Let  the  origin  of  rectangular  coordinates  be  the  pole, 
and  let  the  polar  axis  coincide  with  the  axis  of  x. 

Then  (Fig.  44) 

031=  OP  cos  3I0P, 
PM=  OP  sin  3£0P. 
Or  cc  =  /o  cos  6, 

y=^p  sin  6. 
(ii)  If  the  pole  is  the  point  (h,  k),  we  have 
x  =  h-{-  p  cos  $, 
y  =  k-\-p  sin  0. 
(iii)  If  the  pole  coincides  with  the  origin,  but  the  polar 
axis  OA  makes  the  angle  a  with  the  axis  of  x,  we  obtain 
x  =  p  cos  (^  +  a), 
y  =  p  sin  (^  +  a). 
(iv)    If  the  pole  is  the  point  (h,  k),  and  the  polar  axis 
makes  the  angle  a  with  the  axis  of  x, 

x  =  h-]r  p  cos  (6-\-a), 
y  =  k-\-p  sin  (<9  +  a). 


DIFFERENT  SYSTEMS  OF  COORDINATES.        109 

90.  To  deduce  the  fonnulas  for  finding  the  rectangular 
equation  of  a  curve  from  its  polar  equation. 

From  the  results  in  cases  (i)  and  (ii)  of  §  89  (the  only 
cases  of  importance),  we  readily  obtain 

In  case  (i),   p^  =  x--\-  i/,  tan  6  =  -- 


In  case  (ii),    p^  =  (x  —  hy-\-{y  —  k),^  tan  ^  = 


X 

y—k 


X  —  h 

91.  The  degree  of  an  equation  is  not  altered  by  passing 
from  one  set  of  axes  to  another. 

For,  however  the  axes  may  be  changed,  the  new  equation 
is  always  obtained  by  substituting  for  x  and  y  expressions 
of  the  form 

ax-\-by-\rc   and  a'x  -\-  />'//  -\-  c'. 

These  expressions  are  of  the  first  degree,  and,  therefore,  if 
they  replace  x  and  y  in  the  equation,  the  degree  of  the  equa- 
tion cannot  be  raised.  Neither  can  it  be  lowered  ;  for  if  it 
could  be  lowered,  it  would  be  raised  by  returning  to  the 
original  axes,  and  therefore  to  the  original  equation. 

Exercise  26. 

1.  What  does  the  equation  y"^  —  4x  +  4?/  +  8  =  0  become 
when  the  origin  is  changed  to  the  point  (1,  — 2)  ? 

Transform  the  equation  of  the  circle  (x  —  «)"  -f  (y  —  by 
=  j"^  by  changing  the  origin  : 

2.  To  the  centre  of  the  circle. 

3.  To  the  left-hand  end  of  the  horizontal  diameter. 

4.  To  the  npper  end  of  tlie  vertical  diameter. 

5.  What  does  the  equation  x^-\-  y^^r^  become  if  the  axes 
are  turned  through  the  angle  a  ? 

6.  What  does  the  equation  x^  —  y^  =  a^  become  if  the 
axes  are  turned  through  — 45°  ? 


110  ANALYTIC    GEOMETRY. 

7.  The  equation  of  a  curve  referred  to  rectangular  axes 
is  X  —  xy  —  y^=  0.  Transform  it  to  new  axes,  whose  origin 
is  the  point  ( — 1,  1),  the  new  axis  of  y  bisecting  two  of 
the  angles  formed  by  the  old  axes. 

8.  Change  the  following  equations  to  polar  coordinates, 
taking  the  pole  at  the  origin  and  the  polar  axis  to  coincide 
with  the  axis  of  x  : 

(i)  a;^  -|-  y^  =  a^.      (ii)  x^  —  )f  =  a^. 

9.  Change  the  equation  x^=:4a?/  to  polar  coordinates, 
(i)  taking  the  pole  at  the  origin  ;  (ii)  taking  the  pole  at 
the  point  (a,  0). 

10.  Change  the  following  equations  to  rectangular  coordi- 
nates, the  origin  coinciding  with  the  pole,  and  the  polar  axis 
with  the  axis  of  x  : 

(i)  p  ^=  a,    (ii)  p  =  a  cos  6,    (iii)   p^  cos  26  ==  al 
Transform  the  following  equations  by  changing  the  origin 
to  the  point  given  as  a  new  origin  : 

11.  x-\-i/-\-  2  =  0;  the  new  origin  ( — 2,  0). 

12.  2x  —  5y  —  10  =  0  ;   the  new  origin  (5,  —  2). 

13.  ^x"^ -\- -ixij -\- if  —  5x  —  Qij — 0  =  0;  new  origin  (J, — 4). 

14.  x^  -\-y'^  —  2x  —  4y  =  20  ;  new  origin  (1,  2). 

1 5 .  a;^  —  Q)xy  -\-  if  —  Cj:'  +  2 //  -|-  1  =  0  ;  new  origin  (0,  —  1) . 

16.  Transform  the  equation  x'  —  v/2-|-6  =  0  by  turning 
the  axes  through  45°. 

17.  Transform  the  equation  (x-\-  y  —  2rt)^=:  4a:-^  by  turn- 
ing the  axes  through  45°. 

18.  Transform  the  equation  9^^ — 16//=  144  to  oblique 

axes,  such  that  the  new  axis  of  x  makes  witli  the  old  axis 

3 

of  x  the  negative  angle  tan~'  — -  ;  and  the  new  axis  of  y 

.  .  '-> 

makes  with  the  old  axis  of  x  the  i)0sitive  angle  tan  ~^-. 

4 


DIFFKKKNT    SYSTEMS    OF    C06ui>INATFS.  Ill 

Exercise  27.     (Review^.) 

1.  Find  the   distance   from   the  point  ( — 2b,  b)  to  the 
origin,  the  axes  making  the  angle  60°. 

2.  The  axes  making  the  angle  w,  find  the  distance  from 
the  point  (1,  —  1)  to  the  point  (—  1,  1). 

3.  The  axes  making  the  angle  w,  find  the  distance  from 
the  point  (0,  2)  to  the  point  (3,  0). 

Determine  the   distance   between  the   following  points 
given  by  polar  coordinates  : 

4.  {a,  6)  and  {b,  <j>). 

5.  {a,  6)  and  (a,  —  6). 

6.  {a,  6)  and  (—  a,  —  0). 

7.  (2a,  30°)  and  (a,  60°). 

8.  Show  that  the  polar  coordinates  (p,  6),  ( — ,o,  7r  +  ^), 
( — p,  6  —  tt)  all  represent  the  same  point. 

9.  Transform  the  equation  H.r--\-^.ri/-^4//--\-l2x-\-S}/ 
+  1  =  0  to  the  new  origin  ( —  ^,  —  ^). 

10.  Transform  the  equation  6x^+3^" — 24.r  +  6  =  0  to 
the  new  origin  (2,  0). 

11.  Transform  the  equation --|-^  =  1  by  changing  the 
origin  to  the  point   (  -,  -  )  and  turning  the  axes  through  an 


angle  cb,  such  that  tan  d>  = 

"^  a 

12.  Transform  the  equation  17.r-  —  16.r//  -f- 1 7;/-  =  225  to 
axes  that  bisect  the  axes  of  the  old  system. 

Transform  the  following  rectangular  equations  to  polar 
equations,  the  polar  axis  in  each  case  coinciding  with,  or 
being  parallel  to,  the  axis  of  x,  and  the  pole  being  at  the 
point  whose  coordinates  are  given  : 


112  ANALYTIC    GEOMETRY. 

13.  x^-\-i/=Sax  ;  the  pole  (0,  0). 

14.  X-  -\-  ij-^'6ax;  the  pole  (4a,  0). 

15.  3/-  —  6y  —  5.x  +  9  =  0  ;  the  pole  (| ,  3). 

16.  x^—if  —  Ax-Q>j  —  U  =  0;  the  pole  (2,  —  3). 

17.  ix--{-y-y  =  k\x''—ij')  ;  the  pole  (0,  0). 

Transform  the  following  polar  equations  to  rectangular 
axes,  the  origin  being  at  the  pole  and  the  axis  of  x  coincid- 
ing with  the  polar  axis  : 

18.  p^sm2e  =  2a^ 

19.  p  =  k  sin  2$. 

20.  p(sin  3^  +  cos  3^)  =  5k  sin  $  cos  d. 


21.  Through  what  angle  must  a  set  of  rectangular  axes 
be  turned  in  order  that  the  new  axis  of  x  may  pass  through 
the  point  (5,  7)  ? 

22.  The  rectangular  equation  of  a  straight  line  is  Ax  +  By 
■j-  C=0.  Through  what  angle  must  the  axes  be  turned  in 
order  that 

(i)   the  term  containing  x  may  disappear  ? 
(ii)    the  term  containing  y  may  disappear  ? 

23.  Deduce  the  following  formulas  for  changing  from  one 
set  of  oblique  axes  to  another,  the  origin  remaining  the  same  : 

x'  sin  (o>  —  a)  j^  y'  sin  (w  —  /3) 

sin  0)  sin  w 

x'  sin  a  ,  y'  sin  B 

y=' — ■■ r' — '■ — 

sm  0}         sm  w 

Note.  In  these  formulas  w  denotes  the  angle  formed  by  the  old 
axes,  a  and  ^  those  formed  by  the  positive  directions  of  the  new  axes 
with  the  positive  direction  of  the  old  axis  of  x. 

24.  From  the  formulas  of  I^o.  23  deduce  those  of  §  88, 


CHAPTEE  V. 
THE  PARABOLA. 

The  Equation  of  the  Parabola. 

92.  A  Parabola  is  the  locus  of  a  point  whose  distance 
from  a  fixed  point  is  always  equal  to  its  distance  from  a 
fixed  straight  line. 

The  fixed  point  is  called  the  Focus;  the  fixed  straight 
line,  the  Directrix. 

The  straight  line  through  the  focus  perpendicular  to 
the  directrix  is  called  the  Axis  of  the  parabola. 

The  intersection  of  the  axis  and  the  directrix  is  called 
the  Foot  of  the  axis. 

The  point  in  the  axis  halfway  between  the  focus  and 
the  directrix  is,  from  the  definition,  a  point  of  tlie  curve  ; 
this  point  is  called  the  Vertex  of  the  parabola. 

The  straight  line  joining  any  point  of  the  curve  to  the 
focus  is  called  the  Focal  Radius  of  the  point. 

A  straight  line  passing  through  the  focus  and  limited  by 
the  curve  is  called  a  Focal  Chord. 

The  focal  chord  perpendicular  to  the  axis  is  called  the 
Latus  Rectum  or  Parameter. 

93.  To  construct  a  parabola,  having  given  the  focus  and 
the  directrix. 

I.  Bij  Points.  Let  F  (Fig.  45)  be  the  focus,  CE  the 
directrix.  Draw  the  axis  FD,  and  bisect  FD  in  A  ;  tlien 
A  is  the  vertex  of  the  curve.  At  any  point  M  in  the  axis 
erect  a  perpendicular.      From  F  as  centre,  with  DM  as 


114 


ANALYTIC    (iKOMKTUV. 


radius,  cut  this  perpendicular  in  1*  and  (J ;  then  /*  and  Q 
are  two  points  of  the  curve,  for  7*7^  =  X'iJ/=  distance  of  F 
or  Q  from  CE.  In  the  same  way  we  can  find  as  many 
points  of  the  curve  as  we  please.  After  a  sufficient  number 
of  points  has  been  found,  we  draw  a  continuous  curve 
through  them. 


Fig.  45.  Fig.  46. 

II.  Bij  Motion.  Place  a  ruler  so  that  one  of  its  edges 
shall  coincide  with  the  directrix  DE  (Fig.  46).  Then  place 
a  triangular  ruler  BCE  with  the  edge  CE  against  the 
edge  of  the  first  ruler.  Take  a  string  whose  length  is 
equal  to  BC ;  fasten  one  end  at  B  and  the  other  end  at  F. 
Then  slide  the  ruler  BCE  along  the  directrix,  keeping  the 
string  tightly  pressed  against  the  ruler  by  tlie  point  of  a 
pencil  F.  The  point  F  will  trace  a  parabola  ;  for  during 
the  motion  we  always  have  FF=FC. 

94.  To  find  the  rectangular  equation  oftheparahola,  when 
its  axis  is  taken  as  the  axis  of  x  and  its  vertex  as  the  origin. 

Let  F  (Fig.  45)  be  the  focus,  CE  the  directrix,  DF.Y  the 
axis,  A  the  vertex  and  origin;  also  let  2^)  denote  the  known 
distance  FF. 

Let  F  be  any  point  of  the  curve  ;  then  its  coordinates  are 
A3I=x,     MF  =  y. 


THK    I'AKAHOLA. 


115 


Draw  PC_Lto  CE\  then  by  the  detiuition  of  the  curve 

FF^PC  =  1)M. 
Therefore,  FF' =  ljJr. 

Now  FF''^JIF'+l^'=i/--\-(x—jjy, 

and  lJJl'=^(x-\-py. 

Therefore,  y^  +  (•*'  ~ P) '  =^  (•'■f  +  p)"- 
Whence,  y^=  4.pjc.  [2G] 

This  is  called  the  principal  equation  of  a  parabola. 


95.  Since  ?/  and  jj  in  equation  [2G]  are  positive,  x  must 
always  be  positive;  therefore,  the  curve  lies  wholly  on  the 
positive  side  of  the  axis  of  y. 

An  examination  of  equation  [26]  shows  that  the  curve, 
(i)  passes  through  the  origin,  (ii)  is  symmetrical  with  respect 
to  the  axis  of  x,  (iii)  extends  towards  the  riglit  without 
limit,  (iv)  recedes  from  the  axis  of  x  without  limit. 

96.  Any  point  (h,  />-)  is  outside,  on,  or  inside  the  jiorahola 
jf=^A:px,  arrordin;/  <is  k^  —  -iph  is  positive,  zero,  or  ner/atire. 

Let  Q  be  the  point  (A,  k),  and  let  its  ordinate  meet  the 
curve  in  P. 


116  ANALYTIC    GEOMETRY. 

If  k"^  —  4ph  =  0,  the  point  (A,  k)  satisfies  equation  [26] 
and  therefore  Q  coincides  with  P. 

If  k^  —  4ph  is  positive,  or  k^'^-iph,  then,  since  P3I  = 
Aph,  we  have  QM''>FJI',  or  QM>FM;  hence,  Q  is 
outside  the  curve. 

If  k'^—Aph  is  negative,  we  may  prove  similarly  that  Q 
must  be  inside  the  curve. 

97.  If  X  =-p,  y  =  ±  2/?.  But  these  two  values  of  y  make 
up  the  latus  rectum.     Hence,  the  latus  rectum  =  4:2j. 

Cor.    From  the  equation  y^  =  4px,  it  follows  that 
x:y  =  y:4p; 
that    is,   the    latus   rectum  is  a   third  proportional  to  any 
abscissa  and  its  corresponding  ordinate. 

98.  If  (xi,  yi)  and  (x^,  2/2)  s^re  any  two  points  on  the 
parabola,  we  have 

yj^  =  4:pxi,     y^^  =  ipx2. 
Hence,  yi^  ■  y2^  =  ^i- ^2', 

that  is,  the  squares  of  the  ordinates  of  any  two  points  on  the 
parabola  are  to  each  other  as  their  abscissas. 

99.  To  find  the  points  in  which  the  straight  line  y=^mx 
+  G  meets  the  parabola  y"^  =  4px. 

Eegarding  these  equations  as  simultaneous,  and  eliminat- 
ing X,  we  have . 

Whence,  y=^±^_^  jp-mc^  ^2) 

-^        7n        7n   \      p  ^  ^ 

From  (2)  it  follows  that  y  =  vix-\-c  has  two  distinct,  two 
coincident,  or  no  points  in  common  with  g-^^ipx,  according 
as  ^  —  mc  >,  =,  or  <  0. 

CoR.    If  pj  —  mc  =  0,  or  c^=p-^ni,  y  =  mx-\-c  will  be  a 

tangent;  that  is,   y  =  i7ix-\-—  (3) 

is  a  tangent  to  y'^^Apx  in  terras  of  its  slope. 


THE    PAKAUOLA.  117 

Exercise  28. 

1.  Show  that  the  distance  of  any  point  of  the  parabola 
y^  =:  ^px,  from  the  focus  is  equal  to  p  -\-  x. 

2.  Find  the  equation  of  a  parabola,  taking  as  axes  the 
axis  of  the  curve  and  the  directrix. 

3.  Find  the  equation  of  a  parabola,  taking  the  axis  of 
the  curve  as  the  axis  of  x  and  the  focus  as  the  origin. 

4.  The  distance  from  the  focus  of  a  parabola  to  the 
directrix  =  5.     Write  its  equation, 

(i)  if  the  origin  is  taken  at  the  vertex, 
(ii)  if  the  origin  is  taken  at  the  focus, 
(iii)  if  the  axis  and  directrix  are  taken  as  axes. 

5.  The  distance  from  the  focus  to  the  vertex  of  a  parab- 
ola is  4.  Write  its  equations  for  the  three  cases  enumerated 
in  No.  4. 

6.  For  what  point  of  the  parabola  ?/-:=18x  is  the  ordi- 
nate equal  to  three  times  the  abscissa? 

7.  Find  the  latus  rectum  of  tlie  following  parabolas : 

y'^=^Qx,      f/'^=lox,      hif^ax. 

Find  the  points  common  to  the  following  ])arabolas  and 
straight  lines : 

8.  if  =  ^x,     3.C  —  7y -f  oO  =  0. 
^    9.  f  =  ?>x,     ic  — 4//+12  =  0. 

10.  if  =  '^.x,     a.- =  9,     a=0,     a- =  —  2. 

11.  ?/-  =  4x,     ?/  =  6,     y  =  — 8. 

12.  What  must  be  the  value  of  ^;  in  order  that  the  parab- 
ola ij'^  =  'ipx  may  pass  through  the  point  (9,  —  12)  ? 


118  ANALYTIC    GEOMETRY. 

13.  For  what  point  of  the  pavabohx  i/''  =  o2x  is  the  ordi- 
nate equal  to  4  times  the  abscissa  ? 

14.  The  equation  of  a  pavaboLa  is  i/-  =  Sx.  What  is  the 
equation  of  (i)  its  axis,  (ii)  its  directrix,  (iii)  its  hitus 
rectum,  (iv)  a  focal  chord  tli rough  the  point  whose  ab- 
scissa =8,  (t)  a  chord  })assing  through  the  vertex  and  the 
negative  end  of  the  latus  rectum  ? 

15.  The  equation  of  a  parabola  is  ?/^  =  16a;.  Find  the 
equation  of  (i)  a  chord  through  the  points  whose  abscissas 
are  4  and  9,  and  ordinates  positive ;  (ii)  the  circle  passing 
through  the  vertex  and  the  ends  of  the  latus  rectum, 

16.  If  the  distance  of  a  point  from  the  focus  of  the 
parabola  if  =  4:j)x  is  equal  to  the  latus  rectum,  what  is  the 
abscissa  of  the  point  ? 

17.  In  the  parabola  y^  =  A2jx  an  equilateral  triangle  is 
inscribed  so  that  one  vertex  is  at  the  origin.  What  is  the 
length  of  one  of  its  sides  ? 

18.  A  double  ordinate  of  a  parabola  =  8/).  Prove  that 
straight  lines  drawn  from  its  ends  to  the  vertex  are  jperpen- 
dicular  to  each  other. 

Explain  how  to  construct  a  parabola,  having  given  : 

19.  The  directrix  and  the  vertex. 

20.  The  focus  and  the  vertex. 

21.  The  axis,  vertex,  and  latus  rectum. 

^^22.    The  axis,  vertex,  and  a  point  of  the  curve. 

23.  The  axis,  focus,  and  latus  rectum. 

24.  Determine,  as  regards  size  and  position,  the  relations 
of  tlie  following  parabolas  : 

(i)  y^z=4jjx,  (ii)  f  =  —  4:px,  (iii)  x-  =  4:j)i/,  (iv)  x^  =  —  4rp//. 


the  pakabola,  119 

Tangents  and  Nokmals. 

1 00.  To  find  the  equation  of  the  tangent  and  of  the  normal 
to  tJte  parabola  y^='^2^x  at  any  j^oint  (a*!,  ?/i). 

Let  (xi,  yi),  (x.2,  ?/o)  be  any  two  points  on  the  parabola; 
then  the  equation  of  the  secant  through  them  is 

Ju         «/  1  i/^2  ~~^  OC-t 

Since  (.Ti,  ?/i)  and  (.r^,  //o)  are  on  tlie  cvirve  y-  =  ^jjx,  we  have 
2/2  —  !h  _     k^     . 


a-2  —  a-i       V/2  +  Vi 
By  substituting  in  (1),  the  et^uation  of  the  secant  becomes 

rzjh^_J^.  (2) 

x  —  x^      y^  +  yi 

Now,  if  (xo,  jfo)  is  made  to  coincide  witli  (x^,  //i),  (2)  be- 
comes the  equation  of  tlie  tangent  ix\>(xi,y^.  Putting //a:=yi, 
clearing  of  fractions,  and  remembering  that  y^^-^px^,  Ave 
obtain  as  the  equation  of  the  tangent  at  (xi,y^, 

yr!/=22){.r  +  x,).  [27] 

The  normal  passes  through  (x^,  y^,  and  is  perpendicular 
to  the  tangent;  hence,  its  equation  is,  by  [27]  and  §415, 

101.    If  in  [27]  we  put  //  =  0,  we  obtain 

.<•  =  — .Ti,  or  TA  =  AM(Y\g.  48). 
Therefore,  tlie  snbtangent  Is  bisected  at  the  vertex. 
If  in  [28]  we  put  y=^0,  we  obtain 

x  =  x^  +  2p,  or  X  —  xi  [=  J/xV]  =  2/;  (Fig.  48). 
Hence,  the  stibnoi-inal  is  constant  and  e</ual  to  the  scini- 
latus  rectum. 


120 


ANALYTIC    GEOMETRY. 


Cor.  These  properties  furnish  simple  methods  for  draw- 
ing tangents  to  the  parabola.  Thus,  to  draw  a  tangent  to 
the  parabola  at  P  (Fig.  48),  draw  the  ordinate  PM,  lay  off 
AT=AM,  and  draw  PT,  which  will  be  the  tangent  at  P 
by  §  101.  Or  lay  off  MN=  FD,  and  draw  PN;  then  PT 
perpendicular  to  PN  at  P  will  be  the  tangent  at  P. 


Fig.  48. 

102.    In  the  triangle  FPT(Yig.  48)  we  have 
FT=  FA  +  A  T=p  +  X, 
FP  =  PC=  DM=  DA  -I-  AM=^p  +  x. 

Therefore,       FT=FP. 

Hence,  the  angle 

FPT=  PTF=  TPC,  or 

The  tangent  to  a  parahoht  at  a mj  point  makes  equal  aiigles 
7vith  the  axis  of  the  curve  and  the  focal  radius  to  the  point  of 
contact. 


THE    PARABOLA.  121 

Iizercise  29. 

1.  The  normal  to  a  parabola  at  any  point  bisects  the 
angle  between  the  focal  radius  and  the  line  drawn  through 
the  point  parallel  to  the  axis. 

Note.  The  use  of  parabolic  reflectors  depends  on  this  property. 
A  ray  of  light  issuing  from  the  focus  and  falling  on  the  reflector  is 
reflected  in  a  line  parallel  to  the  axis  of  the  reflector. 

2.  Explain  how  to  draw  a  tangent  and  a  normal  to  a 
given  parabola  at  a  given  point. 

3.  Prove  that  FC  (Fig.  48)  is  perpendicular  to  FT. 

4.  Prove  that    the    tangent    y=^mx-\-~   touches    the 

parabola  if  =  Apx  at  the  point  [  ^'  — 

5.  Prove  that  the  equation  of  a  normal  to  the  parabola 
y^^=^px  in  terms  of  its  slope  is  rj  =  mx  —  mp  (2  +  m?). 

6.  What  are  the  equations  of  a  tangent  and  a  normal 
to  the  parabola  y'^=^ox,  that  pass  through  the  point  whose 
abscissa  is  20  and  ordinate  positive  ? 

7.  What  are  the  equations  of  the  tangents  and  the  nor- 
mals to  the  parabola  ^^=:  12a*,  drawn  through  thg  ends  of  the 
latus  rectum?    Find  the  area  of  the  iigure  they  enclose. 

8.  Through  the  point  on  the  parabola  y^^=  Idx  whose 
abscissa  is  7  and  ordinate  positive  a  tangent  and  a  normal 
are  drawn.  Find  the  lengths  of  the  tangent,  the  normal, 
the  subtangent,  and  the  subnormal. 

9.  A  tangent  to  the  parabola  if  =  20x  makes  with  the 
axis  of  X  an  angle  of  45°.     Determine  the  point  of  contact. 

10.  Show  that  the  focus  i^(Fig.  48)  is  equidistant  from 
the  points  F,  T,  N.  What  easy  way  of  drawing  a  tangent 
and  a  normal  is  sugiiested  by  this  theorem  ? 


122  ANALYTIC    GEOMETRY. 

11.  If  F  is  the  focus  of  a  parabola,  and  Q,  R  denote  the 
points  in  which  a  tangent  cuts  the  directrix  ?ind  the  latus 
rectum  produced,  prove  that  FQ=FM. 

12.  Prove  that  the  tangents  drawn  through  the  ends  of 
the  hitus  rectum  are  perpendicular  to  each  other. 

13.  Find  the  distances  of  the  vertex  and  the  focus  from 

the  tangent  v/=  mx  -\ 

14.  Find  the  point  of  intersection  of  the  tangents  to  the 
parabola  y^  =  ^px  at  the  points  (x^,  3/1),  (x^,  2/2)- 

15.  A  tangent  to  the  parabola  if^=^px  cuts  equal  inter- 
cepts on  the  axes.  Wliat  is  its  equation  ?  What  is  the 
point  of  contact?     What  is  the  value  of  each  intercept? 

16.  Through  what  point  in  the  axis  of  x  must  tangents  to 
the  parabola  if  =  4,jjx  be  drawn  in  order  that  they  may  form 
with  the  tangent,  through  the  vertex,  an  equilateral  triangle  ? 

17.  For  what  point  of  the  parabola  f  =  4:jjx  is  tlie  nor- 
mal equal  to  twice  the  subtangent? 

18.  For  what  point  of  the  parabola  if=^^px  is  the  nor- 
mal equal  to  the  difference  between  tlie  subtangent  and  the 
subnormal  ? 

19.  Find  tlie  equation  of  the  tangent  to  the  parabola 
y^^^x  parallel  to  the  straight  line  ox  —  2_y  +  7  =  0.  Also 
find  the  point  of  contact. 

20.  Find  the  equation  of  the  straight  line  that  touches 
the  parabola  y"=  12.«  and  makes  an  angle  of  45°  with  the 
line  y  =  ^x  —  4.     Also  find  the  point  of  contact. 

21.  Find  the  equation  of  a  straight  line  that  touches 
the  parabola  y^  =  16.r  and  passes  through  the  point  ( —  4, 8). 

22.  If  a  normal  to  a  parabola  for  the  point  P  meets  the 
curve  again  in  the  point  Q,  find  tlie  length  of  PQ. 


THE    PARABOLA.  123 

23.  Trove  by  the  secuiit  method  that  the  e(|uation  of  a  tan- 
gent to  the  parabola  if^=^\px  —  4^r,  at  the  point  (iCi,  v/i)  is 

VxU  ^  2^^  (■^  +  -^i)  —  ^f- 

24.  Find  the  equations  of  the  tangents  and  normals  to 
the  parabola  if  —  '^x  —  G^  —  63^0,  for  the  points  whose 
common  abscissa  =;  —  1. 

25.  What  are  the  general  equations  of  tangents  to  the 
following  parabolas  : 

(i)  y-  =  —  ^px  ?     (ii )  .7--  ^  ^pii  ?     (iii)  x-  =  —  Ajjt/  ? 

Exercise  30.     (Revie^Ar.) 

Note.  If  not  otherwise  specified,  tlie  axis  of  the  parabola  and  tlie 
tangent  at  the  vertex  are  to  be  assumed  as  the  axes  of  coiirdinates. 

What  is  the  equation  of  a  parabola  : 

1.  If  the  axis  and  directrix  are  taken  as  axes,  and  the 
focus  is  the  point  (12,  0)  ? 

2.  If  the  axis  and  tangent  at  the  vertex  are  the  two 
axes,  and  (25,  20)  is  a  point  on  the  curve  ? 

3.  If  the  same  axes  are  taken,  and  the  focus  is  tlie 
point  (—4^,  0)? 

4.  If  the  axis  is  parallel  to  the  axis  of  x,  the  vertex  is 
the  point  (5,  — 3),  and  the  latus  rectum  =  5^? 

5.  If  the  axis  is  tlie  line  ?/  =  —  7,  the  abscissa  of  the 
vertex  =  3,  and  one  point  is  (4,  —  5)  ? 

6.  If  the  curve  passes  through  the  points  (0,  0),  (3,  2), 
(3,-2)? 

7.  If  the  curve  passes  through  tlie  I'.oints  (0,  0),  (3,  2), 
(-3,2)? 


124  ANALYTIC    GEOMETRY. 

8.  What  is  the  latus  rectum  of  the  parahohi  2//^  =  ou'  ? 
What  is  the  equation  of  its  directrix,  and  of  the  focal 
chords  passing  through  the  points  whose  abscissa  :=  6  ? 

9.  Describe  the  change  of  form  which  the  parabola  if  == 
4px  undergoes  as  we  suppose  p  to  diminish  without  limit. 

10.  Find  the  intercepts  of  the  parabola 

i/-{-ix  —  6ij  —  16  =  0. 

11.  One  vertex  of  an  equilateral  triangle  coincides  with 
the  focus,  and  the  others  lie  on  the  parabola  ^/^  =  4^0:.  Find 
the  length  of  one  side. 

12.  The  latus  rectum  of  a  parabola  =  8;  find 

(i)  Equation  of  a  tangent  through  its  positive  end. 
(ii)  Distance  from  the  focus  to  this  tangent, 
(iii)  Equation  of  the  normal  at  this  point. 

13.  What  is  the  equation  of  the  chord  passing  through 
the  two  points  of  the  parabola  y^  =  8x  for  which  x  =  2, 
y  >  0,  and  x  =  18,  y  <  0  ? 

14.  Find  the  equation  of  the  chord  of  the  parabola 
fz=4,px  that  is  bisected  at  a  given  point  (x^,  y^). 

15.  In  what  points  does  the  line  x-]ry  ^=12  meet  the 
parabola  ^/^  +  2x  — 12?/  +  16  =  0  ? 

16.  In  what  points  does  the  line  3?/^  2a; +  8  meet  the 
parabola  y'^  —  4cX~Sy -\-24:  =  0  ? 

17.  Find  the  equations  of  tangents  from  the  origin  to 
the  parabola  (y  —  by=^  A'p(x  —  a). 

18.  Describe  the  position  of  the  parabola  y^  +  2a:;  +  4  =  0 
with  respect  to  the  axes,  and  determine  its  latus  rectum, 
vertex,  focus,  and  directrix. 

19.  What  is  the  distance  from  the  origin  to  a  normal 
drawn  through  the  end  of  the  latus  rectum  of  the  parabola 

y'^=^4^a(x  —  a)  ? 


THE    PARABOLA.  125 

Find  the  equation  of  the  parabola  : 

20.  If  the  equation  of  a  tangent  is  4y  =  3x  —  12. 

21.  If  a  focal  radius  =  10,  and  its  equation  is  3^  =  4a;  —  8. 

22.  If  for  a  point  of  the  curve  the  focal  radius  =  r,  and 
the  length  of  the  tangent  =  ^. 

23.  If  for  a  point  of  the  curve  the  focal  radius  =  r,  and 
the  length  of  the  normal  =  n. 

24.  If  for  a  point  of  tlie  curve  tlie  length  of  the  tangent 
=  t,  and  the  length  of  the  normal  =  n. 

25.  If  for  a  point  of  the  curve  the  focal  radius  =  r,  and 
the  subtangent  =:  s. 

26.  Two  parabolas  have  the  same  vertex,  and  the  same 
latus  rectum  4p,  but  their  axes  are  J_  to  each  other.  What 
is  the  length  of  their  common  chord  ? 

27.  Through  the  three  points  of  the  parabola  7/^  =  12a', 
whose  ordinates  are  2,  3,  6,  tangents  are  drawn.  Show 
that  the  circle  circumscribed  about  the  triangle  formed  by 
the  tangents  passes  through  the  focus. 

28.  A  tangent  to  the  parabola  7f=^px  makes  the  angle 
30°  with  the  axis  of  x.     At  what  point  does  it  cut  the  axis? 

29.  For  what  point  of  tlie  parabola  if^Apx  is  the  length 
of  the  tangent  equal  to  4  times  the  abscissa  of  the  point  of 
contact  ? 

30.  The  product  of  the  tangent  and  normal  is  equal  to 
twice  the  square  of  the  ordinate  of  the  point  of  contact. 
Find  the  point  of  contact  and  the  inclination  of  the  tangent 
to  the  axis  of  x. 

31.  Two  tangents  to  a  parabola  are  perpendicular  to 
each  other.     Find  the  product  of  their  subtangents. 


126  ANALYTIC    GEOMETKY. 

32.  Prove  that  the  circle  described  on  a  focal  radius  as 
diameter  touches  the  tangent  drawn  through  the  vertex. 

33.  Trove  that  the  circle  described  on  a  focal  chord  as 
diameter  tpuches  tiie  directrix. 

Find  the  locus  of  the  middle  points  : 

34.  Of  all  the  ordinatos  of  a  parabola. 

35.  Of  all  the  focal  radii. 

36.  Of  all  the  focal  chords. 

37.  Of  all  chords  passing  through  the  vertex. 

38.  Of  all  chords  that  meet  at  the  foot  of  the  axis. 

Two  tangents  to  the  parabola  if=^^px  make  the  angles 
6,  6'  with  the  axis  of  x ;  find  the  locus  of  their  intersection  : 

39.  It  cot  6 -\- cot  6' =  k.  41.    If  tan  ^  tan  ^' =  7c. 

40.  If  cot  e  —  cot  e'  =  k.  42.    If  sin  6  sin  e'^k. 

43.  Find  the  locus  of  the  centre  of  a  circle  that  passes 
through  a  given  point  and  touches  a  given  straight  line. 

SUPPLEMENTARY   PPtOPOSITIONS. 

1 03.  Two  distmct,  two  coincident,  or  no  real  tunyents  can  he 
drawn  to  a.  parabola  from  any  point  (Ji,  k),  accordiny  as  the 
jjoint  is  without,  on,  or  icit/iin  the  curve. 

Let  the  tangent  y  =  mx  +  —  pass  through  the  point  (h,  k) 

P 
then,  k  =  mh  -\-  — , 

m 

or  h7n'-  —  km  -|- ^v  =  0. 


Whence,  m  = ^ —  • 


THK    I'AKAI'.OT.A. 


127 


These  values  of  m  arc  real  ami  unequal,  real  and  equal, 
or  imaginary,  according  as  /.-"•^  —  4yy //>,=,  or  <0;  that  is, 
according  as  (Ji,  k)  is  without,  on,  or  within  the  parabola  ; 
hence  the  proposition  (§  90). 

104.  To  find  the  equation  of  the  chord  of  contact  of  two 
tangents  drawn  from  any  external  point  {h,k)  to  the  parabola 

Let  (xi.  iji)  and  {x^,  yi)  be  the  points  of  contact ;  then 
the  equations  of  the  tangents  are 

yiij  =  'lp{x-\-x^. 
Since  (/i,  k)  is  in  both  these  lines,  we  have 

ky^  =  ^p{x,^h),  (1) 

ky,  =  2p{x^^h).  (2) 

From  equations  (1)  and  (2)  we  see  that  both  tlie  points 

(xi,  2/1)  and  (j-2,  y^  lie  in  the  straight  line  whose  equation  is 

ky  =  2p{x-^h).  (3) 

Hence,  (3)  is  the  equation  required. 

105.  To  find  the  equation  ofth  e  pola  r  of  the  pole  (h,  k)  u'ith 
regard  to  the  parabola  y'^  =  4yw;. 

Let  P  be  the  fixed  point  (A,  k), 
PQE  one  position  of  the  revolv- 
ing chord,  and  let  the  tangents  at 
Q  and  R  intersect  in  Piixi,  y^) ;  it 
is  required  to  find  the  locus  of 
Pi,  as  the  chord  turns  about  P. 

Since  PH  is  the  chord  of  con- 
tact of  tangents  drawn  from  the 
point  Pi(xi,  yi),  its  equation  is 
(§  104) 


yiy=^j'(->-  +  ^'i)- 


Fig.  49. 


(1) 


128 


ANALYTIC    GEOMETRY. 


Since  (1)  passes  through  (Ji,  Ic)  we  have 

y,k  =  2p{h  +  x,).  (2) 

But  (xi,  ?/i)  is  any  point  on  the  required  locus,  and  by 
(2)  its  coordinates  satisfy  the  equation 

ky  =  2p(x  +  h).  (3) 

Hence,  (3)  is  the  required  equation,  and  the  polar  is  a 
straight  line. 

Cor.  When  the  pole  (Ji,  k)  is  on  the  curve,  the  polar  is 
evidently  a  tangent  at  (A,  k) ;  when  the  pole  (A,  k)  is  with- 
out the  curve,  the  polar  is  the  chord  of  contact  of  tangents 
from  (A,  k).  Thus  the  tangent  and  chord  of  contact  are  par- 
ticular cases  of  the  polar. 

The  Proposition  of  §  74  may  be  proved  for  poles  and 
polars  with  respect  to  a  parabola. 

106.  To  find  the  locus  of  the  middle  points  of  parallel 
chords  in  the  jiarahola  if'  =  4/>x. 


NT.,-^^^ 

^ 

>-7 

> 

\           / 

; 

'A 

Fig.  50. 


Let  any  one  of  the  chords  PQ  (Fig.  50)  be  y^=mx-\-c, 
and  let  it  meet  the  curve  in  the  points  (x^,  i/i),  {x^,  y^)- 


THE    PARABOLA.  129 

Then  (§  100),  vi=  -^'     •  (1) 

Let  M  (x,  y)  be  the  middle  point  of  PQ  ;  tlien  2^  = 
2/1  +  2/2-     By  substitution  in  (1)  we  obtain 

7H  —  —  or  y  =  -^'  (2) 

y         ^       m  ^  ^ 

a  relation  that  holds  true  for  all  the  parallel  chords,  because 
m  is  the  same  for  all  the  chords.  The  required  locus,  there- 
fore, is  represented  by  (2),  and  is  a  straight  line- parallel  to 
the  axis  of  x,  and  called  a  diameter  of  the  parabola.    Hence, 

Every  diameter  of  a  parabola  is  a  straight  line  parallel  to 
its  axis. 

Every  straight  Ihie  parallel  to  the  axis  is  a  diameter  ;  for 

2» 
m,  and,  therefore  -^,  may  have  any  value  whatever. 
m 

107.  Let  the  diameter  through  Jf  meet  the  curve  at  8, 
and  conceive  the  straight  line  P^  to  move  parallel  to  itself 
till  P  and  Q  coincide  at  8 ;  then  the  straight  line  becomes 
the  tangent  at  S  ;  therefore. 

The  tangent  draivn  through  the  extremity  of  a  diameter 
is  parallel  to  the  chords  of  that  diameter. 

108.  From  the  focus  i^draw  FC  ±  to  PQ,  and  let  EC 
meet  the  directrix  in  the  point  C.  If  6  denotes  the  angle 
which  the  chord  PQ  makes  with  the  axis  of  x,  it  easily 
follows  that  DCF^d  ;  then  we  have 

CD  =  FD cot  0  =  ^-^ 
m 

hence,  by  (2)  §  106, 

The  perpendicular  from  the  focus  to  a  chord  meets  the 
diameter  of  the  chord  in  the  directrix. 

Moreover,  since  DS  (Fig.  50)  is  parallel  to  QP,  the  per- 
pendicular from  the  focus  to  a  tangent  and  the  diameter 
through  the  point  of  contact  meet  in  the  directrix. 


130  ANALYTIC    GEOMETRY. 

109.  Let  the  tangents  drawn  throngh  /'  and  Q  meet  in 
the  point  T.      Regarding  their  equations, 

y^ij  =  2p  {x  +  .T2), 
as  simultaneous,  we  obtain  for  tlie  value  of  the  ordinate  of  T 

y  —  -J-LJ h>  _  _L.     Hence, 

y  2  —  1/1  I'l 

Tangents  draum  throufjh  the  ends  of  a  chord  meet  m  the 
diameter  of  the  chord. 

110.  To  find  the  locus  of  the  foot  of  a  per2yendicular  from 
tlie  focus  to  a  tangent. 

Let  the  equation  of  a  tangent  be 

ni 
Then  the  equation  of  the  perpendicular  will  be 

X        p 

y  = 1 — 

ni      VI 
Since  these  two  lines  have  the  same  intercept  on  the  axis 
of  y,  they  meet  in  that  axis  ;  hence,  the  tangent  tlirough  tlie 
vertex  is  the  required  locus. 

111.  Since  FP  =  PC  (Fig.  48)  and  angle  EPC  =  EPF, 
therefore  the  tangent  P  T  is  perpendicular  to  FC  at  its  middle 
point,  and  every  point  in  it  is  equally  distant  from  i^and  C. 

112.  Tangents  at  right  angles  intersect  in  the  directrix. 
Let  the  equation  of  one  tangent  be 

y  =  inx  -\ (1) 

Then  the  equation  of  the  other  is 

y  =  —  --7np.  (2) 

Subtracting  (2)  from  (1)  we  obtain  for  tlieir  common  point 


THE    PARABOLA. 


131 


(x-\-p)(vi-\-    -)=0. 


But  m  -\ cannot  be  zero  ;  lience,  x  -\-p  =  0,  or  x  =  — p, 

which  is  the  equation  of  the  directrix. 

113.  The  polar  of  the  focus  (p,  0)  is 

0  =  2p  (x  -{-p),  or  X  =  —p. 

Hence,  the  polar  of  the  focus  is  the  directrix,  or  tangents 
at  the  ends  of  a  focal  chord  intersect  in  the  directrix. 

CoR.  From  this  result  and  §  112  it  follows  that  tanyents 
throufjh  the  ends  of  a  focal  chord  intersect  at  right  angles. 

114.  To  find  the  equation  of  a  parabola  referred  to  any 
diameter  and  the  tangent  thnnigh  its  extremitij  as  axes. 

Transform  the  equation  ]/  =  4:px  to  the  diameter  SX' 
(Fig.  51)  and  the  tangent  through  S  as  new  axes.  Let  m 
be  the  slope  of  the  tangent,  9  the  angle  which  the  tangent 
makes  with  the  diameter  ;  then  m  =  tan  6. 


First  transform  to  new  parallel  axes  through  S. 
Now,  by  §  106,  BS=2p^m;  hence,  from  y"  =  Apx  we 
obtain  AB=^p-^vr.     Therefore,  the  now  equation  is 


..■+4 

III' 


or 


my-  -\-  4py  =  4j>iii.v. 


(1) 


132 


ANALYTIC    GKOMETRT. 


Now  retain  the  axis  of  x^  and  turn  the  axis  of  y  till  it  coin- 
cides with  the  tangent  at  S\  then  for  any  point  P  we  have 

The  old  X  =  SR.  The  new  x  =  SN. 

The  old  ?/  =:  RP.  The  new  y  =  NP. 

Now  it  is  easily  seen  from  Fig.  52  that 

>S'^  =  SN  +  NP  cos  e,         RP  =  NP  sin  6. 


Therefore,  equation  (1)  is  transformed  to  the  new  system 
by  writing  x -\r  y  cos  6  in  place  of  x,  and  y  sin  6  in  place  of 
y.  Making  this  substitution,  remembering  that  m  =  tan  6, 
and  reducing,  we  obtain 

an  equation  of  the  same  form  as  y'^=^^px. 

Join  S  to  the  focus  F,  and  denote  FS  by  p' ;  then 

^        ^  ?>i''  m^  sin^  ^ 

Therefore,  equation  (2)  may  be  more  simply  written 

y-  =  ^p'x,  (3) 

where  p'  is  the  distance  of  the  origin  from  the.  focus.  It  is 
easy  to  see  that  this  equation  includes  the  case  when  the 
axes  are  the  axis  of  the  curve  and  the  tangent  at  the  vertex. 
The  quantity  4/»'  is  called  the  Parameter  of  the  diameter 
passing  through  >S^.  When  the  diameter  is  tlie  axis  of  the 
curve,  Ap'  is  called  the  Principal  Parameter. 


THE    PARABOLA. 


133 


Cor.    Let  the  equation  of  a  parabola  referred  to  any 

diameter,  and  the  tangent  at  the  end  of  that  diameter  as 

axes,  be  y^=-\p'x.     Since  the  investigations  in  §§  99,  100 

hold  good  whether  the  axes  are  at  right  angles  or  not,  it 

r/ 
follows  immediately  that  the  straight  line  y=^mx-\-  —  will 

touch  the  parabola  for  all  values  of  m,  and  that  the  equation 
of  the  tangent  at  any  point  (xi,  y^  is  yxy=^'2,xj'{x-\-x^. 

115.    To  find  the  2)olar  equation  of  the  jmrabola,  the  focus 
being  the  j^ole. 

Let  P  be  any  point  (p,  6)  of  the 
curve  ;  then 

p  =  FF  =  XP  =  DM=  2p  +  FM 

■=2p-\- p  cos  Q. 

.      -       2y> 


' '  ^     1  —  cose 

Discussion  of  [29] 


[29] 


Fig.  53. 


Since  cos  Q  cannot  exceed  + 1,  p  is  positive  for  all  values 
of  ^. 

If  ^^0,  cos  ^==1,  and  p=^v:. 

This  shows  that  the  axis  of  the  parabola  does  not  cut 
the  curve  to  the  right  of  the  focus. 


If  ^r=|,r,  cos  ^^  0 
If  ^  =  TT,  cos  ^  =  —  1 
If  ^=3^,  cos  ^=  0 
If  ^  =  27r,  cos  ^=      1 


p  =  2p  =  semi-latus  rectum. 
P=   p  =  FA. 
p  =  2p  =  FR'. 

p  =  X. 

As  6  increases  from  zero  to  tt,  p  decreases  from  x  to  p. 
As  6  increases  from  n  to  2ir,  p  increases  from  p  to  x. 


134  ANALYTIC    GEOMETRY. 

Exercise  31. 

1.  Given  a  parabola,  to  draw  its  axis  (§106). 

2.  Prove  that  the  perpendicular  dropped  from  any  point 
of  the  directrix  to  the  polar  of  the  point  passes  through 
the  focus. 

3.  To  find  by  construction  the  pole  of  a  focal  chord. 

4.  Prove  that  through  any  point  tJiree  normals  can  be 
drawn  to  a  parabola. 

5.  Tangents  are  drawn  through  the  ends  of  a  chord. 
Prove  that  the  part  of  the  corresponding  diameter  con- 
tained between  the  chord  and  the  intersection  of  the 
tangents  is  bisected  by  the  curve. 

6.  Focal  radii  are  drawn  to  two  points  of  a  parabola, 
and  tangents  are  then  drawn  through  these  poiuts.  Prove 
that  the  angle  between  the  tangents  is  equal  to  half  the 
angle  between  the  focal  lines. 

7.  Show  that  if  the  vertex  is  tnken  as  pole,  the  polar 
equation  of  a  parabola  is 

A})  cos  6 

^~    s\n-B 

8.  Explain  how  tangents  to  a  ]>arabola  may  be  drawn 
from  an  exterior  point  (§  102). 

9.  Having  given  a  ])arabola,  how  would  you  fiud  its 
axis,  directrix,  focus,  and  latus  rectum? 

10.  From  the  point  ( — 2,  5)  tangents  are  drawn  to  the 
parabola //-  =  6,/-.  What  is  the  equation  of  the  chord  of 
contact? 

11.  The  general  equation  of  a  system  of  parallel  chords 
in  the  parabola  lif  =  25a-  is  4.r  —  7//  +  ^•'  =  0.  What  is  the 
equation  of  the  corresponding  diameter? 


THE    PARABOLA.  135 

12.  lu  the  parabola  if  =  Vox,  \vh;t,t  is  the  equation  of  the 
ordinates  of  the  diameter  y  -f~  H  =  0  ? 

13.  In  the  parabola  /  =  (jj',  wliat  chord  is  Ijisccted  at 
the  point  (4,  3)  ? 

14.  Given  the  parabola  //  =  4/?:r ;  find  the  equation  of 
the  chord  that  passes  through  tlie  vertex  and  is  bisected 
by  the  diameter  y  =  a.    How  can  this  chord  be  constructed  ? 

15.  The  latus  rectum  of  a  parabola  =  16.  Y/hat  is  the 
equation  of  the  curve  if  a  diameter  at  the  distance  12  from 
the  focus,  and  the  tangent  through  its  extremity,  are  taken 
as  axes? 

16.  Show  that  the  equation  of  that  chord  of  the  i)arab- 
ola  if^^-ipx  which  is  bisected  at  the  point  (//,  k)  is 

l-{l,-k)=^2p{x-h). 

17.  Prove  that  the  parameter  of  any  diameter  is  equal 
to  the  focal  chord  of  that  diameter. 

18.  Prove  that  the  locus  of  ij-  —  8//  —  G.'-  +  -8=:0  is  a 
parabola  wliose  axis  is  parallel  to  the  axis  of  x ;  and  deter- 
mine the  latus  rectum,  the  vertex,  the  focus,  the  axis,  and 
the  directrix. 

19.  Prove  that  in  general  the  locus  of  ?/-+-''•'■  + ^'//  + 
C=0  is  a  parabola  Avhose  axis  is  parallel  to  the  axis  of  ./• ; 
and  determine  its  latus  rectum,  vertex,  and  axis. 

20.  Prove  that  in  general  the  \oqx\^  oi  x^ -\- Ax -\- B i/ -\- 
C  =  Q  is  a  parabola  whose  axis  is  parallel  to  the  axis  of  //; 
and  determine  its  latus  rectum,  vertex,  and  axis. 

21.  Find  the  locus  of  the  centres  of  circles  that  touch  a 
given  circle  and  also  a  given  straight  line. 

22.  The  area  and  base  of- a  triangle  being  given,  find  the 
locus  of  the  intersection  of  perpendiculars  dropped  from 
the  ends  of  the  base  to  the  opposite  sides. 


CHAPTER  VI. 
THE    ELLIPSE. 

Simple  Properties  of  the  Ellipse. 

116.  The  Ellipse  is  the  locus  of  a  point,  the  sum  of  whose 
distances  from  two  fixed  points  is  constant. 

The  fixed  points  are  called  Foci;  and  the  distance  from 
any  point  of  the  curve  to  a  focus  is  called  a  Focal  Radius. 

The  constant  sum  is  denoted  by  2a,  and  the  distance 
between  the  foci  by  2c. 

The  fraction  -  is  called  the  Eccentricity,  and  is  repre- 
sented by  the  letter  e.     Therefore,  c  =  ae. 

In  the  ellipse  a  >  c ;  that  is,  e  <C1. 

If  a=-c,  the  locus  is  simply  the  limited  straight  line 
joining  the  foci. 

If  a  <  c,  from  the  definition  it  is  clear  that  there  is  no 
locus. 

117.  To  construct  an  ellipse,  having  giiien  the  foci  and  the 
constant  snm  2a. 

I.  B>/  Motion.  Fix  pins  in  the  paper  at  the  foci.  Tie  a 
string  to  them,  making  the  length  of  the  string  exactly 
equal  to  2a.  Then  press  a  pencil  against  the  string  so  as 
to  make  it  tense,  and  move  the  pencil,  keeping  the  string 
constantly  stretched.  The  point  of  the  pencil  will  trace 
the  required  ellipse  ;  for  in  every  position  the  sum  of  the 
distances  from  the  point  of  the  pencil  to  the  foci  is  equal 
to  the  lensrth  of  the  strinsr. 


THE    ELLIPSE. 


137 


II.    Bij  Points.     Let  F,  F'  be  the  loci;  then  FF'  =  2c. 

Bisect  FF'  at  0,  and  from  0  lay  off  OA=OA'  =  a. 

Then  A' A  =  2a,  F'A'  =  FA, 

A'F-\-A'F'  =  AF-{-FA   =2a, 
AF'+AF  =AF'-\-F'A'  =  2a. 

Therefore,  A  and  A'  are  points  of  the  curve. 

Between  i^and  F'  mark  any  point  X;  then  describe  two 
arcs,  one  with  F  as  centre  and  AX  as  radius,  the  other  with 
F'  as  centre  and  AX  as  radius  ;  the  intersections  P,  Q  of 


these  arcs  are  points  of  the  curve.     By  merely  interchang- 
ing the  radii,  two  more  points,  E,  S,  may  be  found. 

After  a  sufficient  number  of  points  has  been  obtained, 
draw  a  continuous  curve  through  them. 

118.  The  line  AA'  is  the  Transverse  or  Major  Axis, 
A,  A'  the  Vertices,  and  0  the  Centre  of  the  curve. 

The  line  BB\  perpendicular  to  the  major  axis  at  0,  is 
the  Conjugate  or  Minor  Axis;  its  length  is  denoted  by  2b. 

Show  that  B  and  B'  are  equidistant  from  the  foci,  tliat 
BF=a,  that  BO  =  h,  and  that  a- =  Ir -\- r  ^  Ir -[- a-e\ 


138  ANALYTIC    GKOMETRY. 

119.    To  find  the  equation  of  the  ell'qjse,  having  given,  the 
foci  and  the  constant  sum  2a. 

B 


Take  the  line  AA'  (Fig.  55),  passing  through  the  foci,  as 
the  axis  of  x,  and  the  point  0,  halfway  between  the  foci, 
as  origin.  Let  F  be  any  point  (x,  y)  of  the  curve,  and  let 
r,  r'  denote  the  focal  radii  of  P.  Then  from  the  definition 
of  the  curve,  and  from  the  right  triangles  F'PM,  FPM, 


r"=>/+{c  +  xy, 

(1) 

?•-=  if-\-  (c — xy. 

(2) 

By  addition, 

r''J^v^=^2(x'  +  y^  +  c'). 

(3) 

By  subtraction. 

r'-  —  r  =  4:cx. 

(4) 

But 

r'-^r  =  2a. 

(5) 

By  division, 

2ex 
a 

(6) 

By  subtraction, 

ex 

r^=a =  \a  —  ex\. 

a       '-            -^ 

(7) 

By  addition, 

r'  =  a  +  --  =  [a  +  ex~\. 

(8) 

Substitute  in  (3)  remembering  that  h'=^a~  —  c'  (§  118). 
Then  U'x''-\-a-if  ^a'U', 

or  ^!  +  ^  =  1.  [30] 


THE    ELLIPSE.  139 

Cor.  If  the  transverse  axis  is  on  the  axis  of  y,  and  tlie 
conjugate  on  the  axis  of  x,  the  equation  of  the  elli})se  is 

^+-A=i.  (10) 

120.  To  trace  the  form  of  ike  ellipse  from  its  equxtion. 

The  intercepts  on  the  axis  of  x  are  +  a  and  —  a\  on  tlie 
axis  oi  y,  -\-b  and  —  h. 

Only  the  squares  of  tlie  variables  x  and  y  appear  in  the 
equation;  hence,  if  it  is  satisfied  by  a  point  {x,y),  it  will 
also  be  satisfied  by  the  points  (x,  —  y),  ( —  x,  y),  (—  x,  —  //). 
Therefore,  we  infer  that 

(i)   The  curve  is  symmetrical  vnfh  respect  to  the  axis  of  x. 
(ii)    Tlie  curve  is  symmetrical  with  respect  to  the  axis  of  y. 

(iii)  The  curve  is  symmetrical  with  resperf  to  the  centre  O, 
which  bisects  every  chord  passing/  through  it.  This  explains 
why  0  is  called  the  centre. 

\  2  /    \  2 


Since  the  sum  of  I  -  |  and  [  t\   is  1,  neither  of  these 
\aj  \bj 

squares  can  exceed  1 ;  therefore,  the  maximum  value  of  .r  is  +  a, 
and  the  minimum  value  —  o,  while  the  corresponding  values 
of  y  are  -f-  b  and  —  b.  Therefore,  the  curve  is  wholly  con- 
tained within  the  rectangle  whose  sides  are  x^±a,  y  =  ±b. 

121,  To  trace  the  chanyes  in  the  form  of  the  ellijjse  wlicu 
the  semi-axes  are  supposed  to  change. 

Let  a  be  regarded  as  a  constant,  and  b  as  a  variable. 

(i)  Sujipose  b  to  increase.  Then  c  decreases  (since  c'^  = 
a^  —  i'^),  e  decreases,  the  foci  approach  the  centre,  and  the 
ellipse  approaches  the  circle. 

(ii)  Let  b  =  a.  Then  c  =  0,e  =  0,  the  foci  coincide  with 
the  centre,  the  ellipse  becomes  a  circle  of  radius  (/,  and 
equation  [30]  becomes  the  equation  of  the  circle, 


140  ANALYTIC    GEOMETRY. 

(iii)  If  we  suppose  b  to  decrease  to  0  (a  remaining  con- 
stant), c  will  increase  to  a,  e  will  increase  to  1,  while  the 
curve  will  approach,  and  linally  coincide  with,  the  major 
axis,  its  equation  at  the  same  time  becoming  i/  =  0. 

122.  Let  {Xi,i/i)  and  (^'2,2/2)  be  any  two  points  on  the 
ellipse  i^x^  +  a-^^  =  a-(5''^;  then  we  have 

Dividing  and  factoring,  we  have 

2/1^ :  2/2^^ : :  (a  —  Xj)  {a  +  x^) :  (a  —  x^)  {a  +  a^g). 

This  is,  the  squares  of  any  two  ordinates  of  the  ellipse  are 
to  each  other  as  the  products  of  the  segments  into  which  they 
divide  the  major  axis. 

123.  It  follows  from  §  119  that  a  point  (Ji,  k)  is  on  the 
ellipse  represented  by  the  equation  [30],  provided 

-.  +  7-2-1=0. 
a^      V 

It  may  be  shown  by  reasoning  similar  to  that  employed 

in  §  96  that  the  point  (A,  li)  is  outside  or  inside  the  curve, 

h\  k""      ,  . 
according  as  — 2  ryi  —  1  is  positive  or  negative. 

1 24.  If  A,  B,  C  all  have  the  same  sign,  every  equation 
of  the  form  ^^2+^^2_C'  (1) 
may  be  reduced  to  the  form 


x"  .  y      .        X    ,  y      ^ 
a^      W  W      a? 

Hence,  every  equation  of  the  form  of  (1)  represents  an 

ellipse  whose  semi-axes  are  a/—  and  -y  „*    The  transverse 

axis  lies  on  the  axis  of  x  or  the  axis  of  y,  according  as  A 
is  less  than  or  greater  than  B. 


THE    ELLIPSE. 


141 


125.    The  chord  passing  through  either  focus  perpendicu- 
lar to  the  majoraxis  is  called  the  Latus  Rectum  or  Parameter. 
To  find  its  length,  put  a;  =  c  in  the.  equation  of  the  ellipse. 
b""  ,  „       „.       b*  b'' 


Then, 


r=^(s^-<^)  =  ^ 


Therefore,  the  latus  rectum 


21/ _  r4i/ 

a        |_2a 


Forming  a  proportion  from  this  equation,  we  have 
2a  :  2b  : :  2b  :  latus  rectum  ; 
that  is,  the  lathis  rectum  is  a  third  li^'ojiortional  to  the  major 
and  minor  axes. 

126.    The  circle  having  for  diameter  the  major  axis  of 
the  ellipse  is  called  the  Auxiliary  Circle ;  its  equation  is 
cc^  -f-  y"  =  al 
The  circle  having  for  diameter  the  minor  axis  is  called 
the  Minor  Auxiliary  Circle ;  its  equation  is 
a-  +  u  =  ti-. 


If  P  (Fig.  56)  is  any  point  of  an  ellipse,  and  the  ordinate 
MP  produced  meets  the  auxiliary  circle  in  Q,  the  point  Q 
is  said  to  correspond  to  tlie  point  J^. 

The  angle  QOM  is  called  the  Eccentric  Angle  of  the  point 
P,  and  is  denoted  by  the  letter  (^. 


142 


ANALYTIC    GKOMETRY. 


127.  Lft  ij,  ij  represent  the  orcliuates  of  points  in  an 

ellipse  and  the  auxiliary  circle  respectively,  corresponding 

to  the  same  abscissa  x.    Tlieu  from  the  equations  of  the  two 

curves  we  have  b    .—^ j      ,       ,     i—^ :j- 

y  =  rb  -  V  «  —  X  ,    y  —±  'si  a  —  X-. 

Whence,  y:y'^b:a, 

or,  the  ordinates  of  the  eUijise  and  the  auxiliary  circle,  corre- 
sponding to  a  common  abscissa,  are  to  each  other  in  the  con- 
stant ratio  of  the  semi-minor  a7id  semi-major  axes  of  the  ellipse. 

128.  The  principle  of  §  127  furnishes  the  following  easy 
method  of  constructing  an  ellipse  by  points  when  its  axes 
are  given: 


Construct  both  the  major  and  minor  auxiliary  circles 
(Fig.  57);  draw  any  radius  cutting  the  circles  in  B  and  Q; 
through  Q  draw  a  line  parallel  to  BO,  and  through  R  draw 
a  line  parallel  to  OA;  the  intersection  P  of  these  lines  is  a 
point  on  the  ellipse.     For  we  have 

MP:MQ  =  OB:OQ, 
or  MP  \%J  —  b\a. 

From  this  proportion  and  that  in  §  127,  we  have  MP  =  y; 
hence,  P  is  a  point  on  the  ellipse.  In  like  manner  any  num- 
ber of  points  may  be  found. 


THE    ELLIPSE. 


143 


(1) 


Cor.  From  Fig.  57,  we  have 

ic=  0M=  OQ  cos  <f)^a  cos  <f>, 
y=MP=  0N=  OR  sill  <i,  =  b  sin  </,. 

Equations  (1),  which  express  the  coordinates  of  any  point 
of  the  ellipse  in  terms  of  its  eccentric  angle,  may  be  used  as 
the  equations  of  the  ellij^se.  To  obtain  from  them  the  com- 
mon equation,  we  have 


CC  7/ 

-  =  cos  d>,  and  7  =  sin  <i. 
ah 

Therefore,    —-\-~^  cos^  <i>  -}-  sin-  <i  =  1. 
a^      b- 


129.    To  find  the  area  of  an  eUijise. 

Divide  the  semi-major  axis  OA'  (Fig.  58)  into  any  number 
of  equal  parts,  through  any  two  adjacent  points  of  division 
M,  N  erect  ordinates,  and  let  the  ordinate  through  M  meet 


the  ellipse  in  P  and  the  auxiliary  circle  in  Q.  Through  P,  Q 
draw  parallels  to  the  axis  of  a-,  meeting  the  other  ordinate 
in  B,  S,  respectively.     Then  (§  127) 

area  of  rectangle  MPRN_MP _h 
area  of  rectangle  MQSN      MQ     a 


144  ANALYTIC    GEOMETRY. 

A  similar  proportion  liolds  true  for  every  corresponding 
pair  of  rectangles. 

Therefore,  by  the  Theory  of  Proportion, 

sum  of  rectangles  in  ellipse b 

sum  of  rectangles  in  circle        a 

This  relation  holds  true  however  great  the  number  of  rec- 
tangles. The  greater  their  number,  the  nearer  does  the  sum 
of  their  areas  approach  the  area  of  the  elliptic  quadrant  in 
one  case,  and  the  circular  quadrant  in  the  other.  In  other 
words,  these  two  quadrants  are  the  limits  of  the  sums  of 
the  two  series  of  rectangles.  Therefore,  by  the  fundamental 
theorem  of  limits, 

area  of  elliptic  quadrant h 

area  of  circular  quadrant      a 
Multiplying  both  terms  of  this  ratio  by  4, 

area  of  the  ellipse h 

area  of  the  circle        a 

But  the  area  of  the  circle  =  ira^ ;  therefore, 

area  of  the  ellipse  =  -aah.  [31] 

Exercise  32. 

What  are  a,  b,  c,  and  e  in  the  ellipse  whose  equation  is; 
^  .  ^^ 
25^16 
2.   x^+2if  =  2? 

4.  Ax''-\-Bi/-  =  l? 

5.  Find  the  latus  rectum  of  the  ellipse  3a;^  +  7y^  =  18. 

6.  Find  the  eccentricity  of  an  ellipse  if  its  latus  rectum 
is  equal  to  one-half  its  minor  axis. 


THE    ELLIPSK.  145 

What  is  the  equation  of  an  ellipse  if  : 

7.  The  axes  are  12  and  8  ? 

8.  Major  axis  =  2G,  distance  between  foci  =  24  ? 

9.  Sum  of  axes  =  54,  distance  between  foci  =  18  ? 

10.  Latus  rectum,  =  ^5*,  eccentricity  =  5  ? 

11.  Minor  axis  =  10,  distance  from  focus  to  vertex  =  1? 

12.  The  curve  passes  through  (1,  4)  and  ( — 6,  1)  ? 

13.  Major  axis  =  20,  minor  axis  =  distance  between  foci  ? 

14.  Sum  of  the  focal  radii  of  a  point  in  the  curve  =  3 
times  the  distance  between  the  foci  ? 

15.  Prove  that  the  semi-minor  axis  is  a  mean  propor- 
tional between  the  segments  of  the  major  axis  made  by 
one  of  the  foci. 

16.  What  is  the  ratio  of  the  two  axes  if  the  centre  and 
foci  divide  the  major  axis  into  four  equal  parts? 

17.  For  what  point  of  an  ellipse  is  the  abscissa  equal  to 
the  ordinate  ? 

Find  the  intersections  of  the  loci  : 

18.  3x^  +  6/  =  11  and  y  =  x  +  1. 

19.  2x^  +  3y'  =  14  and  if  =  4.r. 

20.  a;2  +  7/  =  16  and  a-^  +  //  =  10. 

21.  The  ordinates  of  the  circle  x^-\-  i/-  =  7^  are  bisected  ; 
find  the  locus  of  the  points  of  bisection. 

22.  A  straight  line  AB  so  moves  that  the  points  A  and 
B  always  touch  two  fixed  perpendicular  straight  lines. 
Show  that  any  point  F  in  AB  describes  an  ellipse,  and 
find  its  equation. 


146  ANALYTIC    GEOMETKV. 

23.  What  is  the  locus  of  J.r-  -j-  />'//'  =  C  when  C  is  zero  ? 
When  is  this  locus  imaginary  ? 

24.  Prove  that  the  al)scissas  of  the  ellipse  Ir^x"^ -\-a'^if 
=  11%"^  are  to  the  corresponding  abscissas  of  the  minor 
auxiliary  circle,  x^-\-if^=U^,  SiS  a  :  b. 

25.  Construct  an  ellipse  by  the  method  of  §  128. 

26.  Construct  an  ellipse,  having  given  c  and  b. 

27.  Construct  tlie  axes  of  an  ellipse,  having  given  the 
foci  and  one  point  of  the  curve. 

28.  Construct  the  minor  axis  and  foci,  having  given  the 
major  axis  (in  magnitude  and  position)  and  one  point  of 
the  ellipse. 

29.  A  square  is  inscribed  in  the  ellipse 

a^^  b' 
Find  the  equations  of  the  sides  and  the  area  of  the  square. 


Tangents  and  Normals. 

130.    To  find  the  equations  of  a  tangent  and  of  <i  normal 
to  an  ellipse,  harin;/  f/iren  the  point  of  contact  (.Tj,  i/i). 

Taking  the  equation  of  the  ellipse, 

b-.r^  -j-  a'^//-  =  a'^b'^, 
and  the  equation  of  the  straight  line  through  (.Ti,  iji)  and 
(•^'2,  y^), 

V  —  Vx  _  Vi—yx 

— > 

.r  —  .7'i      .Tg  —  .r  1 
and  proceeding,  as  in  §  G4,  we  obtain  as  the  equation  of  a 
secant  through  (a^i,  y^  and  (xj,  v/o) 

y  —  V\  ^  _  l>\^x  +  a-o) 
a-— a-i  «^(/yi  +  Z/2) 


THE    ELLIPSE.  147 

Now  make  x^^x^,  ij„=zy^-^  then  the  chord  becomes  a  tan- 
gent, and 

x—xj^  (r{!/i  +  7/2) 

y  ~  Vi        ^^^^'i 

becomes '—  = — > 

X  —  Xi  c^  y\ 

which  reduces  to    ^+'Jj!/_  =  1,  ri^n 

From  the  equation  above  it  appears  tliat  the  value  of  the 
slope  of  the  tangent,  in  terms  of  the  coordinates  of  the 
point  of  contact,  is 

b'^Xi 
a% 

The  normal  is  perpendicular  to  the  tangent,  and  passes 
through  (xi,  ?/i)  ;  therefore,  its  equation  is  easily  found  (by 
the  method  of  §  46)  to  be 

i/-!/i  =  |!^;(^-^i).  [33] 

131.    To  find  the  suhtangent  and  subnoDnal. 

Making  y  =  0  in  [32]  and  [33],  and  tlien  solving  the 
equations  for  x,  we  obtain  : 

Intercept  of  tangent  on  axis  of  a-  =  — 

('"^ 
Intercept  of  normal  on  axis  of  x  =:  —  .r,  =  e^x^. 

Whence,  the  values  of  the  subtangent  and  the  subnormal 
(defined  as  in  §  63)  are  easily  found  to  be  as  follows : 


Siibtang-eiit  = 

[34] 

Subiiorinal  = 

a- 

[:ib^ 

148 


ANALYTIC    GEOMETRY. 


132.  If  tangents  to  ellipses  having  a  common  major  axis 
are  drawn  at  points  having  a  common  abscissa,  they  will 
meet  on  the  axis  of  x. 

For  in  all  these  ellipses  the  values  of  a  -and.  x  are  con- 
stant, and  therefore  (by  §  131)  the  tangents  all  cut  the 
same  intercept  from  the  axis  of  x. 

Y 

Q 


Fig.  59. 

133.  The  normal  at  any  point  of  an  ellipse  bisects  the 
angle  formed  by  the  focal  radii. 

The  values  of  the  focal  radii  for  the  point  P  (Fig.  59) 
were  found,  in  §  119  to  be 

PF=a  —  ex^,     PF'=a-\-  exy. 

If  the  normal  through  P  meets  the  axis  of  x  in  N,  ON 
=  e'^Xi  (§  131)  ;  and,  therefore, 

NF  =c  —  e^Xi  =  ae  —  e^.r^  =  e(a  —  ex). 
NF'  ^c-\-  e^Xi  =  ae  -\-  e-x^  =  e(a  +  ex). 

Therefore,    NF  :  NF'  =  PF  :  PF', 
or  the  normal  divides  the  side  FF'  of  the  A  PFF'  into 
two  parts  proportional  to  the  other  two  sides.     Therefore 
(by  Geometry),  Z  FPN=  Z  FPN. 

The  tangent  PT,  being  perpendicular  to  the  normal, 
must  bisect  the  angle  FPG,  formed  by  one  focal  radius 
with  the  otlier  produced. 


THE    ELLIPSE. 


149 


134.  To  draw  a  taiujent  and  a  "lornud  tliroiujh  a  (jlven 
point  of  an  ellipse. 

I.  Let  P  (Fig.  60)  be  the  given  point.  Describe  the  aux- 
iliary circle,  draw  the  ordinate  MP,  produce  it  to  meet  the 
circle  in  Q,  draw  QT  tangent  to  the  circle  and  meeting  the 
axis  of  X  in  T,  and  join  PT;  then  PT  is  a  tangent  to  tlie 
ellipse  (§  132).  Draw  PNl,  to  PT;  PNis  the  normal  at  P. 
C 


II.  Draw  the  focal  radii,  and  bisect  the  angles  between 
them.  The  bisectors  are  the  tangent  and  the  normal  at  the 
point  P  (§  133), 

135.  To  find  the  equation  of  a  tangent  to  an  ellipse  in 
terms  of  its  slope. 

This  problem  may  be  solved  by  finding  under  what  con- 
dition the  straight  line 

y=^mx-]-c  (1) 

will  touch  the  ellipse  b-x-  +  <rir  =  a^b^.  (2) 

Eliminating  y  from  (1)  and  (2),  and  tlien  solving  for  .r. 
we  find  two  values  of  x  : 


—  ma^c  ±  ab\/ni^a'^  -{■  Ir  —  c^ 
""^  m'a'-\-b' 

These  values  will  be  equal  if 

m^a^  -\-b''  —  c-  =  0,  or  c  =  ±  VwVT^'- 


150  ANALYTIC    fJEOMETRY. 

If  the  two  values  of  x  are  equal,  the  two  values  of  y  must 
also  be  equal,  from  equation  (1). 

Therefore,  the  two  points  in  which  the  ellipse  is  cut  by 
the  line  will  coincide  if  c  =  ±  sj vrii- -\- UK 

Hence,  the  straight  line 

y  =  ^nx  ±  ^tti-fi-  +  ly^  C^*^] 

will  touch  the  ellipse  for  all  values  of  m.     . 

Since  either  sign  may  be  given  to  the  radical,  it  follows 
that  two  tangents  liaving  the  same  slo})e  may  be  drawn  to 
an  ellipse. 

136.  To  find  the  locus  of  the  intersertion  of  two  tangents 
to  an  ellipse  tvhich  are  jjerjiendictdar  to  each  other. 

Let  the  equations  of  the  tangents  be 

y^mx  -\-  '\Jm-(r  -\-h^,  (1) 

y  =  m'x  +  Vm'V  +  i^.  (2) 

The  condition  to  be  satisfied  is 

m  m  =  —  1 ,  or  ?»/  = 

m 

If  we  substitute  for  vi'  in  equation  (2)  its  value  in  terms 
of  m,  the  equations  of  the  tangents  may  be  written 

y  —  mx  =  ^Imhv^  +  ^^  (3) 

my  -\-x^  \la--\-  m-/j\  (4) 

The  coordinates,  x  and  y,  of  the  intersection  of  the  tan- 
gents satisfy  both  (3)  and  (4);  but  before  Ave  can  find  the 
constant  relation  between  them  we  must  first  eliminate  the 
variable  7n. 

This  is  most  easily  done  by  adding  the  squares  of  the 
two  equations  ;  the  result  is 

(1  +  ur)x'  +  (1  +  m'-)f  =(1  +  w')  (a^-  +  h% 
or  x^-\-y^^^a"  -\-U-. 

The  required  locus  is  therefore  a  circle.  This  circle  is 
called  the  Director  Circle  of  the  ellipse. 


THE    ELLIPSE.  151 


Exercise  33. 


1.  What  are  the  equations  of  the  tangents  and  normals 
to  the  ellipse  2x^-\-3i/''^^35  at  the  points  whose  abscissa  ==2? 

2.  What  are  the  equations  of  the  tangents  and  normals 
to  the  ellipse  'ix^-\-  9 y^^SG  at  the  points  whose  abscissa=  —  §? 

3.  Find  the  eciuations  of  the  tangent  and  the  normal  to 
the  ellipse  a-^H-4^'==20  at  the  point  of  contact  (2,  2). 
Also  find  the  svibtangent  and  the  subnormal. 

4.  Show  that  the  line  )/  =  x-\-  V^'  touclies  the  ellipse 
2x^  -{-  o//'-  =  1. 

5.  Required  the  condition  whicli  must  be  satisfied  in 

X  II 

order  that  the  straight  line [-  -  =  1  may  touch  the  ellipse 

^+^  =  1. 

6.  In  an  ellipse  the  subtangent  for  the  point  (3,  J^)  is 
—  y-,  the  eccentricity  =  4,  What  is  the  equation  of  the 
ellipse  ? 

7.  What  is  the  equation  of  a  tangent  to  the  ellipse 
9x--|-64//-  =  57G  parallel  to  the  line  2i/^x? 

8.  Find  the  equation  of  a  tangent  to  the  ellipse  3x^-\- 
bif^  =  15  parallel  to  the  line  4a;  —  oy  —  1  =  0. 

9.  In  what  points  do  the  tangents  that  are  equally 
inclined  to  the  axes  touch  the  ellipse  b'\r-  -{-  n^ij^  =  a-lr? 

10.  Tlirough  what  point  of  the  e\\i\)i^e /rx'- -\- tr//- =  (rlr 
must  a  tangent  and  a  normal  be  drawn  in  order  tliat  they 
may  form,  witli  the  axis  of  ./;  as  base,  an  isosceles  triangle? 

11.  Tlirough  a  point  of  the  ellipse  //-.»■- -|- <(-//- =  a-7/-,  anil 
the  corresponding  point  of  the  auxiliary  circle  j'--\-if  =  a-, 
normals  are  drawn.     What  is  tlie  ratio  of  the  subnornuxls? 


152  ANALYTIC    GEOMETRY. 

12.  For  what  points  of  the  ellipse  ^V  +  a}if-  =  a^l?  is  the 
subtangent  equal  numerically  to  the  abscissa  of  the  point 
of  contact  ? 

13.  Find  the  equations  of  tangents  drawn  from  the  point 
(3,  4)  to  the  ellipse  16a;2 +  25/  =  400. 

14.  What  are  the  equations  of  the  tangents  drawn  through 
the  ends  of  thelatera  recta  of  the  ellipse  4a:;^  +  9?/"  =  36a^? 

15.  What  is  the  distance  from  the  centre  of  an  ellipse  to 
a  tangent  making  the  angle  <^  with  the  major  axis? 

16.  What  is  the  area  of  the  triangle  formed  by  the  tan- 
gent in  the  last  problem  and  the  axes  of  coordinates  ? 

17.  From  the  point  where  the  auxiliary  circle  cuts  the 
minor  axis  produced  tangents  are  drawn  to  the  ellipse. 
Find  the  points  of  contact. 

18.  Prove  that  the  tangents  drawn  through  the  ends  of 
a  chord  through  the  centre  are  parallel. 

19.  Find  the  locus  of  the  foot  of  a  perpendicular  dropped 
from  the  focus  to  a  tangent. 

Exercise  34.     (Review.) 

1 .  Given  the  ellipse  mx"  + 100/  =  3600.  Find  the  equa- 
tions and  the  lengths  of  focal  radii  drawn  to  the  point  (8, -'/)• 

2.  Is  the  point  (2,  1)  within  or  without  the  ellipse 
2x^  +  3^2  =  12? 

Find  the  eccentricity  of  an  ellipse  : 

3.  If  the  equation  is  2x'  +  3/  =  12. 

4.  If  the  angle  FBF  =  90"  (see  Fig.  54). 

5.  Show  that  4.t2 -  8x  +  9/y'  -  36?/  +  4  =  0,  or  4(.x  —  1)^ 
4- 9(?/  — 2)2  =  36,  is  an  ellipse  whose  centre  is  (1,  2),  and 
semi-axes  3  and  2. 


THE    ELLIPSE.  153 

Find  the  equations  of  tangents  to  an  ellipse  : 

6.  If  they  make  equal  intercepts  on  the  axes. 

7.  If  they  are  parallel  to  BF  (Fig.  54). 

8.  Which  are  parallel  to  the  line  -  -|-  y  =  1  (a  and  b 
being  the  semi-axes). 

9.  Find  the  equation  of  a  tangent  in  terms  of  the  eccen- 
tric angle  ^  of  the  point  of  contact. 

Find  the  distance  from  the  centre  of  an  ellipse  to  : 

10.  A  tangent  through  the  point  of  contact  (a-j,  ?/i). 

11.  A  tangent  making  the  angle  ^  with  the  axis  of  x. 

12.  In  what  ratio  is  the  abscissa  of  a  point  divided  by 
the  normal  at  that  point? 

13.  At  the  point  (x^,  y^)  of  an  ellipse  a  normal  is  drawn. 
What  is  the  product  of  the  segments  into  which  it  divides 
the  major  axis  ? 

14.  Find  the  length  of  FN  (Fig.  59). 

15.  Determine  the  value  of  the  eccentric  angle  at  the 
end  of  the  latus  rectum. 

Prove  that  the  semi-minor  axis  b  of  an  ellipse  is  a  mean 
proportional  between  : 

16.  The  distances  from  the  foci  to  a  tangent. 

17.  A  normal  and  the  distance  from  the  centre  to  the 
corresponding  tangent. 

Determine  and  describe  the  loci  of  the  following  points  : 

18.  The  middle  point  of  the  portion  of  a  tangent  con- 
tained between  the  tangents  drawn  through  the  vertices. 

19.  The  middle  point  of  a  perpendicular  dropped  from 
a  point  of  a  circle  (x  —  ")*■^"■'/^^ ''"  ^^  ^^^®  '"^-^^^  ^^  U- 


154  ANALYTIC    GEOMETKY. 

20.  The  middle  jtoiiit  of  a  chord  of  the  ellipse  b-y--\- u'l/^ 
^a'-lr  drawn  through  the  positive  end  oi  tlie  minor  axis. 

21.  The  vertex  of  a  triangle  whose  base  2c  and  sum  of 
the  other  sides  2s  are  given. 

22.  The  vertex  of  a  triangle,  having  given  the  base  2c 
and  the  product  k  of  the  tangents  of  the  angles  at  the  base. 

23.  The  symmetrical  point  of  the  right-hand  focus  of  an 
ellipse  with  respect  to  a  tangent. 

SUPPLEMENTARY    PPvOPOSITIONS. 

137.  Two  distinct,  two  coincident,  or  no  tangents  can  he 
drawn  to  an  ellipse  tlirowjli,  auij  point  {It,  k),  according  as 
the  point  is  without,  on,  or  ivithin  the  curve. 

Let  the  tangent  ?/  =  ?/iic+  V?>iV  +  i^  pass  through  the 

point  (A,  k);  then  

k  =  mh  -\-  Vm'^a-  +  ^^ 
or  (Ji"  —  a^)  vi"  —  2hkin  +  k^  —  V  =  0. 


m- 


hk±  \IVK'^a^k^  —  a^lP' 


h:'  —  a^  '  <^^) 

Hence,  there  will  be  two  distinct,  two  coincident,  or  no 

tangents  through  {h,  k),  according  as  h^h^-\- aVc^  —  ci-h^^, 

=,  or<0;  that  is,  according  as  (h,  k)  is  without,  on,  or 

within  the  ellipse. 

138.    To  fijid  the  equation  of  the  chord  of  contact  of  the  two 

tangents  drawn  from  an  cjcternal  point  (Jt,  k)  to  the  ellipse, 

a^~^  V 

Let  the  student  prove,  by  a  course  of  reasoning  similar 

to  that  employed   in   §§71   and   104,   that  the    required 

equation  is 

hx      kg . 


THE    ELLII'SK. 


ir>r, 


139.  To  find  the  equatioii  of  the  iiohir  of  the  pole  (h,  /:), 
with  regard  to  the  ellipse. 

Let  the  student  prove,  by  a  course  of  reasoning  similar 
to  that  employed  in  §§  72  and  105,  that  the  required 
equation  is 

hx      ky 

a^        0- 

CoR.  The  tangent  and  cliord  of  contact  are  particular 
cases  of  the  polar.  The  proposition  of  §  74  holds  true  for 
poles  and  polars  with  regard  to  the  ellipse. 

140.  To  draw  a  tangent  to  an  ellipse  from  a  given  jjoint 
P  outside  the  curve. 


Fig.  CI. 

Suppose  the  problem  solved,  and  let  the  tangent  touch 
the  ellipse  at  Q  (Fig.  (11).  If  F'Q  is  produced  to  G,  making 
QG=QF,  then  A  FQG  is  isosceles;  now  ZFQP  =  Z 
GQP  (§  1.33)  ;  therefore,  PQ  is  per])ondicnlar  to  FG  at  its 
middle  point ;  therefore,  P  is  equidistant  from  /♦"  and  G. 
This  reduces  the  problem  to  determining  the  point  G. 

Since  F'G=^2a,  (7  lies  on  the  circle  with  F'  as  centre 
and  2a  as  radius.  And  G  also  lies  on  the  circle  with  7' 
as  centre  and  PF  as  radius.  Hence  the  constrnction  is 
obvious. 


156  ANALYTIC    GEOMETRY. 

141.  To  Jind  the  locus  of  thi;  middla  points  of  anij  system 
of  jmrallel  chords  in  the  ellipse. 

Let  any  one  of  the  parallel  chords  y  =  mx  +  c  meet  the 

ellipse 

b'x'^  -\-  (Cif  =  a-y^ 

in  the  points  (xi,  yi)  and  {x«.  y^) ;  then,  by  §  130, 

/v-(a-i+a-2) 

If  (a*,  y)  is  the  middle  point,  2a;  =  iCi  +  X2,  2y  =  yi-{-  y^, 
and  (1)  becomes 

Px 

or  y^  —  ——  (2) 

This  relation  holds  true  for  the  middle  points  of  all  the 
chords  ;  therefore,  it  is  the  equation  of  the  locus  required. 

From  (2)  we  see  that  any  straight  line  passing  through 
the  centre  of  an  ellipse  is  a  diameter. 

142.  Let  m!  denote  the  slope  of  the  diameter  of  the 
chords  whose  slope  is  m;  then  from  (2)  of  §  141 

m  = i'  or  mm'  = r.  \  ot  \ 

ma^  a-  ^     -■ 

Thus  [37]  is  the  equation  of  condition  that  the  diameter 
yz=m'x  bisects  all  chords  parallel  to  the  diameter  y  =  mx', 
but  [37]  is  evidently  also  the  equation  of  condition  that 
y  =■  mx  bisects  all  chords  parallel  to  ?/  =  vi'x ;  hence, 

If  one  diameter  bisects  all  chords  parallel  to  another,  the 
second  diameter  bisects  all  chords  parallel  to  the  first. 

Two  such  diameters  are  called  Conjugate  Diameters. 

CoR.  From  [37]  the  slopes  of  two  conjugate  diameters 
must  have  opposite  signs ;  hence,  two  conjugate  diameters 
of  an  ellijyse  lie  on  opposite  sides  of  the  miiior  axis. 


THE    ELLIPSE. 


157 


143.  Let  a  straight  line  cutting  the  ellipse  in  P  and  Q 
move  parallel  to  itself  till  P  and  Q  coincide  with  the  end 
of  the  diameter  bisecting  PQ;  then  the  straight  line  be- 
comes the  tangent  at  the  end  of  the  diameter.     Therefore, 

The  tangents  at  the  extremities  of  any  diameter  are  parallel 
to  the  chords  of  that  diameter ,  and  also  to  its  conjugate  diameter. 

144.  Let  POP'  and  HOP'  (Fig.  62)  be  two  conjugate 
diameters  meeting  the  ellipse  in  the  points  P  (xi,  y^  and 

R  (^2, 3/2)-     The  slope  of  the  tangent  through  P  is ^; 

a  yi 

hence,  the  equation  of  the  diameter  ^0^',  which  is  parallel 
to  this  tangent  (§  143),  is 


h\r^  Xix  .  yiv      „ 

y  = ^x,  or  — -  +  •— -  =  0. 

ayi  a^        ¥ 


(1) 


Fig.   G2. 

Now  R  {x2, 2/2)  is  on  (1),  and  also  on  the  ellipse  ;  hence 
we  have  3.  3.       , , , , 

-J— 2  J_  •ZLZj!  =  0  (0\ 


and 


1. 


Solving  (2)  and  (3)  for  x^  and  y^,  we  obtain 
a  b 

o  a 


(3) 


158  ANALYTIC    GKOMICTRY. 

The  up]ter  signs  give  the  coordinates  of  li,  and  the  lower 
those  of  E'  in  terms  of  a\  and  //i. 

Equation  (2)  is  the  condition  that  must  be  satisfied  by 
the  coordinates  of  the  extremities  of  every  pair  of  conju- 
gate diameters. 

145.  Denoting  the  semi-conjugate  diameters  OP  and  OB 
(Fig.  G2)  by  a'  and  //,  respectively,  we  have 

a'2  —  xf  +  //i"  =  .rj-  +      (((-  —  xA 

=  ^^  +  "^^V^  -=  H'  +  ^'^.\  (1) 

a 

and  h"'  =  jv  + 1/2'  =  J,  Z/r  +  ^^vTi'^  (§  144) 

=  a'  -  j-;'  +  K  x,^  =  a'  —  eW.  (2) 

Adding  (1)  and  (2),  we  have 
a'2  +  ^/2  =  a-  +  i'^. 

That  is,  the  sum  of  the  stianres  of  any  pair  of  semi^cov jugate 
diameters  is  equal  to  the  sum  of  the  squares  of  the  semi-axes. 

Equations  (1)  and  (2)  ex])ress  the  lengths  of  tlie  semi- 
conjugate  diameters  a'  and  //  in  terms  of  a,  b,  and  x^  (the 
abscissa  of  the  extremity  of  a'). 

146.  Let  the  ordinates  of  the  extremities  P,  R  (Fig.  62) 
of  two  conjugate  diameters  meet  the  auxiliary  circle  in  Q,  S, 
respectively,  join  QO  and  SO,  and  denote  Z.  QOX  by  <^, 
Z^  SOX  by  <^'.  Then  the  values  of  the  coordinates  of  P 
and  R  are  (§  127), 

Xi  =  a  cos  <f>,       Xo  =  a  cos  (f>', 

l/i^b  sin  ^,        yi'=b  -sin  <^'. 
Whence,  by  substitution  in  etpiation  (2)  of  §  144,  we  obtain 

cos  <^  cos  ^'-|-sin  <^  sin  <^'=z{). 
Therefore,     cos  (^'  —  ^)  =  0,    or  <^'  —  <^  =  ^tt. 


THK    ELLirSE. 


159 


That  is,  tlie  difference  of  the  eccentric  avrjles  correspond i>ir/ 
to  the  ends  of  two  conjugate  diameters  is  equal  to  a  right  angle. 

CoK.    The  angle  FOR  (Fig.  62)  is  obtuse,  siiu^e  QOS=l_'Tr. 

147.  To  find  the  angle  formed  by  two  conjugate  semi-diam- 
eters, whose  lengths  a',  b'  are  gioen. 

Let  the  semi-diameters  make  the  angles  a,  (3,  respectively, 
with  the  axis  of  .r,  and  let  0  denote  the  required  angle. 
Then  if  (a-j,  ?/i)  and  (x^,  i/o)  are  the  extremities  of  a'  and  b\ 
respectively, 

sin  a 


7/ 


''''f^=b'=7b'- 


cos  a  =  —5      COS  (3 


or., aj/i 

«■  ■         'b'^~Jb'' 

sin  6^  sin  (/S  — a) 

=  sin  )8  cos  a  —  cos  /8  sin  a 
b^x-^-\-a^Ui^        aHr 


aba'b' 


aba'b' 


ah 

a'l^ 


(1) 


CoR.  1.    Clearing  (1)  of  fractions,  we  liave 
a'V  sin  6=^  ah, 
which  shows  that  the  area  of  the  parallelogram  HEKE  is 
equal  to  the  rectangle  L21QN.     (O  ClfJiS=a'b' sin  6.) 
R 
N 


^  jy  w 

Fig.  G3. 

That  is,  the  j>arallelogram  formed  by  tangents  at  the 
extremities  of  any  pair  of  conjugate  diameters  is  equal  to  the 
rectangle  on  the  axes. 


IGO  ANALYTIC    GEOMETRY. 

CoK.  2.  If  CT  (Fig.  G3)  is  perpendicular  to  tlie  tangent 
KK,  then, 

CT^  CD  sin  CDR  =  a'  sin  $  =  j,- 

148.  The  lines  joining  any  point  of  an  ellipse  to  the 
ends  of  any  diameter  are  called  Supplemental  Chords. 

Let  FQ,  F'Q  be  two  supplemental  chords   (Fig.  64). 

Through  the  centre  0  draw  OR  parallel  to  P'Q,  and  meeting 

PQ'm  R;  also  OR'  parallel  to  FQ,  and  meeting  F'Q  in  R'. 

Q 


Fig.  64. 

Since  0  is  the  middle  point  of  FF',  and  OR  is  drawn 
parallel  to  F'Q,  and  OR'  is  drawn  parallel  to  FQ,  R  and  R' 
are  the  middle  points  of  QF,  QF',  respectively.  Therefore, 
OR  will  bisect  all  chords  parallel  to  QF,  and  OR'  will 
bisect  all  chords  parallel  to  QF'.  Hence,  OR,  OR'  are 
conjugate  diameters. 

Therefore,  the  diameters  jxirallel  to  a  pair  of  supplemental 
chords  are  conjugate  diameters. 

CoR.  1.  This  principle  affords  the  following  easy  method 
of  drawing  a  pair  of  conjugate  diameters  which  shall  in- 
clude a  given  angle. 

On  the  transverse  axis  AA'  describe  a  segment  of  a  circle 
which  shall  include  the  sriven  angle.     Let  the  arc  of  this 


THE    ELLirSE.  161 

segment  cut  the  ellipse  in  Q  and  *S';  then  the  diameters 
parallel  to  QA  and  QA',  or  SA  and  SA',  are  conjugate  and 
include  the  required  angle. 

CoR.  2.  If  B  is  the  upper  vertex  of  the  conjugate  axis, 
the  conjugate  diameters  parallel  to  BA  and  BA'  will  evi- 
dently be  equal,  and  will  lie  on  the  diagonals  of  the  rec- 
tangle on  the  axes  of  the  ellipse. 

149.  To  find  the  equation  of  an  ellipse  referred  to  a  pair 
of  conjugate  diameters  as  axes. 

Since  each  of  two  conjugate  diameters  of  the  ellipse 
bisects  the  chords  parallel  to  the  other,  the  curve  is 
(obliquely)  symmetrical  with  respect  to  each  of  the  new 
axes;  hence,  as  the  required  equation  is  of  the  second  degree, 
it  contains  only  the  squares  of  x  and  y,  and  is  of  the  form 
Ax^  +  Bi/=C.  (1) 

The  intercepts  of  the  curve  on  the  new  axes  are  the  semi- 
conjugate  diameters.     Denoting  them  by  a'  and  b',  we  have 

Substituting  these  values  in  (1),  we  obtain 

which  is  the  required  equation  in  terms  of  the  semi-conjugate 
diameters. 

This  equation  has  the  same  form  as  the  equation  referred 
to  the  axes  of  the  curve ;  whence  it  follows  that  formulas 
derived  from  equation  [30],  by  processes  that  do  not  pre- 
suppose the  axes  of  coordinates  to  be  rectangular,  hold  true 
when  we  employ  as  axes  two  conjugate  diameters. 

For  example,  the  equation  of  a  tangent  at  the  point(a:i,2/i), 
referred  to  the  semi-conjugate  diameters  a'  and  b',  is 

a'2  "•"  b''' 


1G2 


ANALYTIC    OKOMKTRY. 


150.     To  consfritrt  the,  polar  of  a  fucus 
Since  the  po 
focus  (ae,  0)  is 


Since  the  polar  of  (h,  k)  is  -7  +  ^  =  1,  tlie  polar  of  the 


aex  =  a;  or  x 


Hence,  ae  :  a  ^=  a  -.  x. 

Therefore,  if  OD  (Fig.  65)  is  taken  so  that 
OF:  0A=^  OA  :  OD, 
and  DC  is  drawn  perpendicular  to   OD,  DC  will  be  the 
polar  of  the  focus  F. 

The  polar  of  a  focus  is  called  a  Directrix  of  the  ellipse. 
Hence,  DC  is  the  directrix  corresponding  to  the  focus  F. 

In  like  manner  we  may  constnict  F'C,  or  the  directrix 
corresponding  to  the  focus  F'. 

Cor.    Let  Q  (x,  ?/)  bo  any  ])oint  on  the  ellipse;  then, 

QS=OD-OM^'^-x  =  '^  =  ^. 

Hence,         e  =  FQ-^QS. 

That  is,  tJie  distances  of  ant/  poivt  on  the  ellipse  from  a 
focus,  and  the  corresj)ondi)if/  directrix,  bear  the  constant  ratio  e. 


THE    ELLIPSE. 


163 


Whence,  the  ellipse  is  often  defined  as  : 

Tlte  locus  of  a  point  ivliicli  moves  so  that  its  distances  from 
a  fixed  2)0 int  and  a  fixed  stvahjlit  line  hear  a  constant  ratio 
less  than  unity. 

151.  To  find  tlie  polar  equation  of  the  ellipse,  the  left- 
hand  focus  being  taken  as  the  jjole. 


Fig.   66. 

Let  F  be  any  point  (p,  0)  of  tlie  ellipse ;  then,  from  equa- 
tion (8)  of  §  119,  we  have 

p  =  a^  ex.  (1) 

Now         X  =  0M=  F'M  —  F'0  =  pcoQe  —  ae. 

Substituting  tins  value  of  x  in  (1),  we  have 
p  =  «  -|-  ep  cos  9  —  ae^. 


Whence,  p 


1  —  ecos6 


[39] 


Cob.  Since  e  <  1,  and  cos  6  cannot  exceed  unity,  p  is 
always  positive. 

),  p^ -^ =  ./  +  ae  =  F'A. 


If 


If 


1  =  0. 


;  Itt,  p=:a(l  —  e'-)  =  F'Ji  =■  senii-latus  rectum. 


164  ANALYTIC    GEOMETRY. 

If  e  =  ^,    p  =  ''-^^^  =  a-ae  =  F'A'. 

^  l-\-e 

If  0  =  ^-,r,  p  =  «(1  —  e^)  =  semi-latus  rectum. 

If  6  =  2ir,p  =  a  +  ae  =  F'A. 

While  6  increases  from  zero  to  tt,  p  decreases  from  a-\-ae 

to  a  —  ae;  and  while  6  increases  from  tt  to  27r,  p  increases 

from  a  —  ae  to  a-\-  ae. 

If  F  is  taken  as  the  pole,  the  polar  equation  is 

a(l  —  e}) 

"       1  +  e  cos  6 

Exercise  35. 

1.  Find  the  area  of  the  ellipse  a:;^  +  4y^  =  16. 

2.  Find  the  distances  of  the  directrices  from  the  centre 
in  No.  1. 

3.  What  is  the  equation  of  the  polar  of  the  point  (5,  7) 
with  respect  to  the  ellipse  4a;^  +  %^  =  36  ? 

4.  Prove  that  a  focal  chord  is  perpendicular  to  the  line 
that  joins  its  pole  to  the  focus.  In  what  line  does  the 
pole  lie  ? 

5.  Find  the  pole  of  the  line  Ax-\-  By -\-  C  =  0  with 
respect  to  the  ellipse  b'^x^  -\-  a? if  =  a%^. 

6.  Each  of  the  two  tangents  that  can  be  drawn  to  an 
ellipse  from  any  point  on  its  directrix  subtends  a  right 
angle  at  the  focus. 

7.  The  two  tangents  that  can  be  drawn  to  an  ellipse 
from  any  external  point  subtend  equal  angles  at  the  focus. 

8.  Find  the  slope  mi  of  a  diameter  if  the  square  of  the 
diameter  is  (i)  an  arithmetic,  (ii)  a  geometric,  (iii)  an  har- 
monic mean  between  the  squares  of  the  axes. 

9.  Given  the  length  21  of  a  diameter,  its  inclination  6  to 
the  axis,  and  the  eccentricity;  find  the  major  and  minor  axes. 


THE    ELLIPSE.  165 

10.  Tangents  at  the  extremities  of  any  chord  intersect 
on  the  diameter  which  bisects  that  chord. 

11.  Tangents  are  drawn  from  (3,  2)  to  the  ellipse 
x^  +  4//-  =  4.  Find  the  equation  of  the  chox-d  of  contact,  and 
of  the  line  that  joins  (3,  2)  to  the  middle  point  of  the  chord. 

12.  Find  the  area  of  the  rectangle  whose  sides  are  the  two 
segments  into  which  a  focal  chord  is  divided  by  the  focus. 

13.  What  is  the  equation  of  a  chord  in  the  ellipse 
13^-  +  11^^143  that  passes  through  (1,  2)  and  is  bisected 
by  the  diameter  ^x  —  2?/  =  0  ? 

14.  In  the  ellipse  9x^  +  36 ?/^  =  324  find  the  equation  of  a 
chord  passing  through  (4,  2)  and  bisected  at  this  point. 

15.  Write  the  equations  of  diameters  conjugate  to  the 
following  lines  : 

X  —  y^O,     a"  -|-  y  =  0,      ax  =  hy,      ay  =  bx. 

16.  Show  that  the  lines  2x  —  y^=0,  x-\-Sy=^0  are  con- 
jugate diameters  in  the  ellipse  2x^  -\-  3y^  ^=  4. 

17.  Find  the  equation  of  a  diameter  parallel  to  tlie 
normal  at  the  point  (xi,  y^),  the  semi-axes  being  a  and  b. . 

18.  The  rectangle  of  the  focal  perpendiculars  upon  any 
tangent  is  constant  and  equal  to  the  square  of  the  semi-minor 
axis. 

19.  The  diagonals  of  the  parallelogram  in  Fig.  63,  §  147, 
are  also  conjugate  diameters. 

20.  The  angle  between  two  semi-conjugate  diameters  is 
a  maximum  when  they  are  equal. 

21.  The  eccentric  angles  corresponding  to  equal  semi- 
conjugate  diameters  are  45°  and  135°. 

22.  The  polar  of  a  point  in  a  diameter  is  parallel  to  tlie 
conjugate  diameter. 


166  ANALYTIC    GEOMKTKV. 

23.  Find  the  ecjuations  of  equal  conjugate  diameters. 

24.  Tlie  length  of  a  semi-diameter  is  I;  find  the  equation 
of  the  conjugate  diameter. 

25.  The  angl(!  between  two  equal  conjugate  diameters 
is  120°  ;  find  the  (;c(u'ntricity  of  the  ellipse. 

26.  Given  a  diameter,  to  construct  the  conjugate  diameter. 

27.  To  draw  a  tangent  to  a  given  ellipse  parallel  to  a 
given  straight  line. 

28.  Given  an  ellipse,  to  find  by  construction  the  centre, 
foci,  and  axes. 

29.  Find  the  rectangular  equation  of  the  ellipse,  taking 
the  origin  at  the  right-hand  vertex. 

30.  Find  the  polar  equation  of  an  ellipse,  taking  as  i)ole 
tlie  right-hand  focus. 

31.  Find  the  polar  equation  of  the  ellipse,  taking  the 
centre  as  pole. 

32.  If  the  centre  of  an  ellipse  is  the  point  (4,  7),  and 
the  major  and  minor  axes  are  14  and  8,  find  its  equation, 
the  axes  being  supposed  parallel  to  the  axes  of  C()()rdinates. 

•  33.    The  equation  of  an  ellipse,  the  origin  being  at  tlie 
left-hand  vertex,  is  25.r--l- 81^ ;=450.r ;  find  the  axes. 

34.  If  the  minor  axis  =  12,  and  the  latus  rectum  =  5, 
what  is  the  equation  of  the  ellipse,  the  origin  being  taken 
at  the  left-hand  vertex  ? 

35.  Find  the  eccentric  angle  <^  corresponding  to  the 
diameter  whose  length  is  2c. 

36.  At  the  intersection  of  the  ellipse  Ji^x^  -\-  c/^  =  aHi'^  and 
the  circle  x--\-/f^(ih  tangents  are  drawn  to  both  curves. 
Find  the  ansrle  between  them. 


THE    ELLIPSE.  167 

37.  How  would  you  draw  a  normal  to  an  ellipse  from 
auy  point  in  the  minor  axis  ? 

38.  Find  the  equation  of  a  chord  that  is  bisected  at 
the  point  (Ji,  k). 

39.  Prove  that  the  length  of  a  line  drawn  from  the 
centre  to  a  tangent,  and  parallel  to  either  focal  radius  of 
the  point  of  contact,  is  equal  to  the  semi-major  axis. 

40.  A  circle  described  on  a  focal  radius  will  touch  the 
auxiliary  circle. 

41.  Find  the  locus  of  the  intersection  of  tangents  drawn 
through  the  ends  of  conjugate  diameters  of  an  ellipse. 

42.  Find  the  locus  of  the  middle  jtoint  of  the  chord 
joining  the  ends  of  two  conjugate  diameters. 

43.  Find  the  locus  of  the  vertex  of  a  triangle  Avhose 
base  is  the  line  joining  the  foci,  and  whose  other  sides  are 
parallel  to  two  conjugated  iameters. 

44.  Show  that  ^iX^  +  y^  +  8a?  —  2y  + 1  =  0  represents  an 
ellipse  ;  find  its  centre  and  axes. 

45.  If  A  and  B  have  like  signs,  sliow  that  tlie  locus  of 
Ax-  + 11  y-  -\-  iKr  -f-  Jiy  +  i''=  0  is  in  general  an  ellipse  whose 
axes  are  parallel  to  the  coordiuate  axes  ;  and  determine  its 
semi-axes. 

46.  Find  the  locus  of  the  centre  of  a  circle  that  passes 
through  the  point  (0,  3)  and  touches  internally  the  circle 
x^  +  1/'  ^  25. 


CHAPTER  VII. 
THE  HYPERBOLA. 

Simple  Properties  of  the  Hyperbola. 

152.  The  Hyperbola  is  the  locus  of  a  point  the  difference 
of  whose  distances  from  two  fixed  points  is  constant. 

The  fixed  points  are  called  the  Foci,  and  a  line  joining 
any  point  of  the  curve  to  a  focus  is  called  a  Focal  Radius. 

The  constant  difference  is  denoted  by  2a,  and  the  dis- 
tance between  the  foci  by  2c. 

The  fraction  -  is  called  the  Eccentricity,  and  is  denoted 
a 

by  the  letter  e.     Therefore,  c-=ae. 

Since  the  difference  of  two  sides  of  a  triangle  is  always 

less  than  the  third  side,  we  must  have  in  the  hyperbola 

2a  <!  2c,  or  a  <C  c,  or  e  >  1. 

153.  To  construct  an  hyperhola,  having  r/iven  the  foci,  and 
the  constant  difference  2a. 

I.  B]/  Motion  (Fig.  67).  Fasten  one  end  of  a  ruler  to 
one  focus  F'  so  that  it  can  turn  freely  about  F'.  To  the 
other  end  fasten  a  string.  Make  the  length  of  the  string 
less  than  that  of  the  ruler  by  2a,  and  fasten  the  free  end 
to  the  focus  F.  Press  the  string  against  the  ruler  by  a 
pencil  point  P,  and  turn  the  ruler  about  F\ 

The  point  P  will  describe  one  branch  of  an  hyperbola. 
The  other  branch  may  be  described  in  the  same  way  by 
interchanging  the  fixed  ends  of  the  ruler  and  the  string. 


THE    HYPEKBOLA. 


169 


II.    By  Points  (Fig.  68).     Let  F,  F'  be  the  foci ;  then 

FF'  =  2c. 
Bisect  FF'  at  0,  and  from  0  lay  off  OA  =  OA'  =  a. 
Then  AA'  =  2a,     FA  =  F'A '. 

AF'—  AF=  AF'  —  A'F'  =  AA'  ^  2a. 

A'F—  A'F'  =  A'F—  AF  =  AA'  =  2a. 
Therefore,  A  and  A'  are  points  of  the  curve. 


Y 

X 

°<l 

/^ 

1    F' 

A' 

1 

0 

A 

^l> 

1 1) 

~--^1k^. 

--V'-- 

A\ 

/Q\ 

^ 

Fig.  67. 


Fig.  68. 


In  FF'  produced  mark  any  point  D ;  then  describe  two 
arcs,  the  first  with  F  as  centre  and  AD  as  radius,  the  second 
with  F'  as  centre  and  A'D  as  radius  ;  the  intersections  P,  Q 
of  these  arcs  are  points  of  the  curve.  By  merely  inter- 
changing the  radii,  two  more  points  B,  S  may  be  found. 

Proceed  in  this  way  till  a  sufficient  number  of  points  has 
been  obtained;  then  draw  a  smooth  curve  through  them. 

Through  0  draw  BB'  J_to  FF'-,  since  the  difference  of 
the  distances  of  every  point  in  the  line  BB'  from  the  foci 
is  0,  therefore  the  curve  cannot  cut  the  line  BB'. 

The  locus  evidently  consists  of  two  entirely  distinct 
parts  or  branches,  symmetrically  placed  with  respect  to  the 
line  BB',  called  the  right-hand  and  the  left-hand  branches. 


170 


ANALYTIC    GEOMETRY. 


1 54.  The  point  (J,  halfwiiy  between  the  foci,  is  the  Centre. 

The  points  A,  A',  where  the  line  passing  through  the 
foci  meets  the  curve,  are  called  the  Vertices. 

The  line  AA'  is  the  Transverse  Axis. 

The  transverse  axis  is  equal  to  the  constant  difference 
'Ja,  and  is  bisected  by  the  centre  (§  153). 


Fig.  CO. 

The  line  BB'  passing  through  0  perpendicular  to  AA' 
does  not  meet  the  curve  (§  153) ;  but  if  B,  B'  are  two  points 
whose  distances  from  the  two  vertices'^,  ^' are  each  equal 
to  c,  then  BB'  is  called  the  Conjugate  Axis,  and  is  denoted 
by  2h. 

Since  AAOB  =  A  A  OB',  OB=OB'=l>;  that  is,  the  con- 
jugate axis  is  bisected  by  the  centre.  ; 

In  the  triangle  AOB,  OA^=a,  OB  =  h,  AB^^i^;^  hence, 
c"  =  a'  -\-  Ir. 

The  chord  passing  through  either  focus  perpendicular  to 
the  transverse  axis  is  the  Latus  Rectum,  or  Parameter. 

Note.  Since  a  and  b  are  equal  to  the  legs  of  a  riglit  triangle,  a 
may  be  greater  than  or  less  than  b;  hence  the  terms  '■'■major''''  and 
^^ minor''''  are  not  appropriate  in  the  hyperbola. 


THIC    HVPKKBOI.A.  17 J 

155.  By  proceeding  as  in  the  case  of  the  ellipse  (§  119), 
using  r'  —  r  =  ±  2(i  instead  of  /•'  +  r  =  2a,  and  substituting 
i^  for  c^  —  a^,  we  obtain  as  the  equation  of  the  hyperbola 

^'-1?=1.  [40] 

Thus  the  equations  of  the  ellipse  and  hyperbola  differ 
only  in  the  sign  of  h^;  that  of  the  ellipse  is  changed  into 
that  of  the  hyperbola  by  substituting  —  V^  for  +  V^.     Hence, 

Any  formula  deduced  front  the  equation  of  the  ellipse  is 
changed  to  the  correspondincj  formula  for  the  hyperbola  by 
merely  changing  -\-b-  to  —  Ir,  or  b  to  b\l —  1. 

The  lengths  r,  r'  of  the  focal  radii  for  any  point  (a*,  //)  are 
r  ^  ±  (ex  —  a  ) ,     r '  =  ±  (ex  +  o  ) , 
in  which  the  upper  signs  hold  for  the  right-hand  branch, 
and  the  lower  for  the  left. 

156.  A  discussion  of  equation  [40]  leads  to  the  follow- 
ing conclusions  : 

(i)  The  curve  cuts  the  axis  of  x  at  the  two  real  points 
{a,  0)  and(— «,  0). 

(ii)  The  curve  does  not  cut  the  axis  of //.   The  imaginary 
intercepts  are  ±  Z*  V  —  1. 

(iii)  No  part  of  the  curve  lies  between  the  straight  lines 
x^-\-a  and  x  =  —  a. 

(iv)  Outside  these  lines  the  curve  extends  without  limit 
both  to  the  right  and  to  the  left. 

(v)  The  greater  the  abscissa,  the  greater  the  ordinate. 

(vi)  The  curve  is  symmetrical  with  respect  to  the  axis 
of  a-. 

(vii)  The  curve  is  symmetrical  with  respect  to  the  axis 
of  y. 

(viii)  Every  chord  that  passes  through  the  centre  is 
bisected  by  the  centre.  This  explains  wli}-  the  })oint  half- 
way between  the  foci  is  called  the  centre. 


172 


ANALYTIC    GEOMETRV. 


157.    An  hyperbola  whose  transverse  and  conjugate  axes 

are  equal  is  called  an  Equilateral  Hyperbola.  Its  equation  is 

oc^-y--a-.  [41] 

The  equilateral  hyperbola  bears  to  the  general  hyperbola 
the  same  relation  that  the  auxiliary  circle  bears  to  the  ellipse. 


Fig.  70. 

158.  The  hyperbola  that  has  BB'  for  transverse  axis, 
and  AA'  for  conjugate  axis,  obviously  holds  the  same  rela- 
tion to  the  axis  of  ?/  that  the  hyperbola  which  has  AA'  for 
transverse  axis  and  BB' for  conjugate  axis  holds  to  the  axis 
of  X. 

Therefore  its  equation  is  found  by  simply  changing  the 
signs  of  a^  and  b^  in  [40],  and  is 


1. 


(1) 


The  two  hyperbolas  are  said  to  be  Conjugate. 


THE    HYPERBOLA.  173 

159.    The  straight  line  z/=?na;,  passing  through  the  centre 

of  the  hyperbola  -^—  -,  =  !>  meets  the  curve  in  two  points, 

the  abscissas  of  which  are 

-\-  ah  —  ah 


\IU^  —  ni^a'  V^^ — m^a^ 

Hence  the  points  will  be  real,  wiarjinavy,  or  situated  at 
infinity,  as  h-  —  m^a?  is  positive,  negative,  or  zero ;  that  is, 

as  rn?  is  less  than,  greater  than,  or  equal  to  —• 

The  same  line,  y  =  mx,  will  meet  the  conjugate  hyperbola 

— r — TZ  =  — 1  in  two  points,  whose  abscissas  are 
a'      ¥ 

-\-  ah,  —  ah 


Hence  these  points  will  be  imaginary,  real,  or  situated  at 

h^ 
infinity,  as  m^  is  less  than,  greater  than,  or  equal  to  —^• 

Whence, 

If  a  straight  line  through  the  centre  meets  an  hyperhola  in 
iraaginary  points,  it  will  meet  the  conjugate  hyjjerhola  in  real 
points,  and  vice  versa. 

160.  An  Asymptote  is  a  straight  line  that  passes  through 
finite  points,  and  meets  a  curve  in  two  points  at  infinity. 

We  see  from  §  159  that  the  hyperbola 

x"^      // 

^  "~  Ij"  ~ 
has  two  real  asymptotes  which  pass  through  the  centre  of 

the  curve,  and  which  have  for  their  equations  y  =  -\--x  and 

b 
X ;  or, 

1^  =  0.  [42] 


y  =  —  -x;  or 


^2 

a2 


174  ANALYTIC    GEOMETRY. 

Exercise  36. 

What  is  the  equation  of  an  hyperbola,  if  : 

1.  Transverse  axis  =  IG,  conjugate  axis  =  14  ? 

2.  Conjugate  axis  =  12,  distance  between  foci  =  13  ? 

3.  Distance  between  foci  =  twice  the  transverse  axis? 

4.  Transverse  axis  =  8,  one  point  is  (10,  25)  ? 

5.  Distance  between  foci  =  2c,  eccentricity  =  V2  ? 

6.  Prove  that  the  latus  rectum  of  an  hyperbola  is  equal 

2lr 

to Also  2a  :  20  ■.•.2b:  latus  rectum. 

a 

7.  The  equation  of  an  hyperbola  is  9x-^  — 16^/ =  144; 
iind  the  axes,  distance  between  the  foci,  eccentricity,  and 
latus  rectum. 

S.  Write  the  equation  of  the  hyperbola  conjugate  to  the 
hyperbola  9x-^— !()//-=  144,  and  find  its  axes,  distance 
between  its  foci,  and  its  latus  rectum. 

9.  If  the  vertex  of  an  hyperbola  bisects  the  distance 
from  the  centre  to  the  focus,  find  the  ratio  of  its  axes. 

10.  I'rove  that  the  point  (.*•,  y)  is  ivitlumt,  on,  or  ivithin 
the  hyperbola,  according  as  —^  —  'j^—^  is  nerjatioe,  zero,  or 

positive. 

11.  Find  the  eccentricity  of  an  equilateral  hyperbola. 

12.  Find  the  points  that  are  common  to  the  hyperbola 
25x-2— 9/=22r),  and  tlie  straight  line  25.^+12^  =  45. 

13.  The  asymptotes  of  an  hyperbola  are  the  diagonals 
of  the  rectangle  CDEG  (Fig.  70,  p.  172). 

14.  Find  the  foci  and  the  asymptotes  of  the  hyperbola 
16a;2  — 9/  =  144. 


THE    HYPERBOLA.  175 

15.  The  asymptotes  of  an  equilateral  hyperbola  are  per- 
pendicular to  each  otlier.  Hence  the  equilateral  hy})erb()la 
is  also  called  the  rectan<juJar  hyperbola. 

16.  Two  conjugate  hyperbolas  have  the  same  asymi)totes. 

17.  Find  tlie  length  of  the  perpendicular  dropped  from 
the  focus  to  an  asymptote. 

18.  Prove  that  the  squares  of  any  two  ordinates  of  an 
hyperbola  are  to  each  other  as  the  products  of  the  segments 
into  which  they  divide  the  transverse  axis  externally. 

Tangekts  and  Normals. 

Note.  The  results  stated  in  the  following  six  sections  may  be 
established  in  the  same  way  as  the  corresponding  propositions  in  the 
ellipse,  or  the  first  five  may  be  obtained  by  §  155. 

161.  The  slope  of  the  tangent  at  (x^,  i/i)  is  —^,  and  its 

a  i/i 

equation  is 

a^_2/LM^l.  [43] 

162.  The  equation  of  the  nnrninl  at  (.Vi,  y{)  is 


O-Jl  I 

[-14] 

63. 

The  subtangent  =^~ ^   the  stdjnormal 

b'ix. 

64. 

The  straight  line  whose  equation  is 

y  =  mx  ±  \lm'a'^  —  b^ 
is  a  tangent  for  all  values  of  m  (§  135). 

165.  The  equation  of  the  director  circle  of  an  hyperbola 

is  a-2_|_y/2  =  ,,2  —  /;2  (^   130). 

166.  The  tangent  and  the  nornaal  at  any  jioint  of  an 
hyperbola  bisect  the  angles  formed  by  tlie  focal  radii  of 
the  point  (§  133). 


176  ANALYTIC    GEOMETRY. 

Exercise  37. 

1.  Find  the  equations  of  tlie  tangent  and  of  the  normal 
at  the  point  (4,  4)  of  the  hyperbola  16a;^  —  %^=  112.  Also 
lind  the  lengths  of  the  subtangent  and  the  subnormal. 

2.  Show  that  in  an  equilateral  hyperbola  the  subnormal 
is  equal  to  the  abscissa  of  the  point  of  contact. 

3.  The  equations  of  the  tangent  and  the  normal  at  a  point 
of  an  equilateral  hyperbola  are  ox  —  4?/  =  9,  4x  ■i-5ij  =  40. 
What  is  the  equation  of  the  hyperbola,  and  what  are  the 
coordinates  of  the  point  of  contact  ? 

4.  For  what  points  of  an  hyperbola,  is  the  subtangent 
equal  to  the  subnormal  ? 

5.  To  draw  a  tangent  and  a  normal  to  an  hyperbola  at 
a  given  point  of  the  curve. 

6.  If  an  ellipse  and  an  hyperbola  have  the  same  foci, 
prove  that  the  tangents  to  the  two  curves  drawn  at  their 
points  of  intersection  are  perpendicular  to  each  other. 

7.  Prove  that  the  asymptotes  of  an  hyperbola  are  the 
limiting  positions  of  tangents  to  the  infinite  branches. 

8.  Prove  that  the  length  of  a  normal  in  an  equilateral 
hyperbola  is  equal  to  the  distance  of  the  point  of  contact 
from  the  centre. 

9.  Find  the  distance  from  the  origin  to  the  tangent 
through  the  end  of  the  latus  rectum  of  the  equilateral 
hyperbola  x^  —  y^=  a^. 

10.  "What  condition  must  be  satisfied  in  order  that  the 

X      ?/  x^      ?/ 

straight  line  — h  -  =  1  may  .touch  the  hyperbola '—  —  7^=!  ? 

11.  When  is  the  director  circle  of  an  hyperbola  imaginary? 

12.  Find  the   locus   of   the  foot  of  the  perpendicular 
dropped  from  the  focus  of  an  hyperbola  to  a  tangent. 


THE    HYPERBOLA.  177 

Exercise  38.     (Review.) 

1.  The  ordinate  through  the  focus  of  an  hyperbola, 
produced,  cuts  the  asymptotes  in  P  and  Q.  Find  PQ  and 
the  distances  of  P  and  Q  from  the  centre. 

2.  In  the  hyperbola  9x^  — 16?/^  =  144  what  are  the 
focal  radii  of  the  points  whose  common  abscissa  is  8  ? 
What  other  points  have  equal  focal  radii  ? 

3.  What  relation  exists  between  the  sum  of  the  focal  radii 
of  a  point  of  an  hyperbola  and  the  abscissa  of  the  point  ? 

4.  Prove  that  in  an  equilateral  hjq^erbola  every  ordinate 
is  a  mean  proportional  between  the  distances  of  its  foot  from 
the  vertices  of  the  curve.  Hence,  find  a  method  of  con- 
structing an  equilateral  hyperbola  when  the  axes  are  given. 

5.  In  an  equilateral  hyperbola  the  distance  of  a  point 
from  the  centre  is  a  mean  proportional  between  its  focal  radii. 

6.  In  an  equilateral  hyperbola  the  bisectors  of  the 
angles  formed  by  lines  drawn  from  the  vertices  to  any 
point  of  the  curve  are  parallel  to  the  asymptotes. 

7.  If  e,  e'  are  the  eccentricities  of  two  conjugate  hyper- 
bolas, 

8.  Through  the  positive  vertex  of  an  hyperbola  a  tangent 
is  drawn.  In  what  points  does  it  cut  the  conjugate  hyperbola? 

9.  The  sum  of  the  reciprocals  of  two  focal  chords  per- 
pendicular to  each  other  is  constant. 

10.  Through  the  foot  of  the  ordinate  of  a  point  in  an 
equilateral  hyperbola  a  tangent  is  drawn  to  the  circle 
described  upon  the  transverse  axis  as  diameter.  What 
relation  exists  between  the  lengths  of  this  tangent  and  the 
ordinate  of  the  point  ? 


178  ANALYTIC    GEOMETRY. 

11.  In  an  equilateral  hyperbola  find  the  equations  of 
tangents  drawn  from  the  positive  end  of  the  conjugate  axis. 

12.  From  what  ])oint  in  the  conjugate  axis  of  an  hyper- 
bola must  tangents  be  drawn  in  order  that  they  may  be 
perpendicular  to  each  other  ? 

13.  What  condition  must  be  satisfied  that  a  square  may 
be  constructed  whose  sides  sliall  be  parallel  to  the  axes  of 
an  hyperbola  and  whose  vertices  shall  lie  on  the  curve  ? 

14.  Find  the  equation  of  the  chord  of  the  hyperbola 
lG.r^  — %-  =  144  that  is  bisected  at  the  point  (12,  3). 

15.  Find  the  equation  of  the  tangent  to  the  hyperbola 
16.X-  — 9//^=144  parallel  to  the  line  y  =  4x  —  3. 

16.  Find  the  product  of  the  two  perpendiculars  let  fall 
from  any  point  of  any  hyperbola  upon  the  asymptotes. 

17.  A  chord  of  an  liyperbola  that  touches  the  conjugate 
hyperbola  is  bisected  at  the  point  of  contact. 

SUPPLEMENTARY   PROPOSITIONS. 

Note.  Many  of  the  following  propositions  are  closely  analogous  to 
propositions  already  establislied  for  tlie  ellipse  ;  hence  the  proofs  are 
omitted,  and  references  given  to  th(i  chapter  on  the  ellipse. 

167.  7'iao  distinct,  two  coincident,  or  no  tangents  can  he 
drawn  to  an  hyperhola  through  any  point  (Ji,  k^,  according  as 
the  point  is  without,  on,  or  within  the  curve  (§  137). 

168.  TJie  equation  of  the  chord  of  contact  of  the  tiro  tan- 
gents draum  from  an  external  point  (Ji,  /.•)  to  the  hypterhola 

'-,-f=l,     is^-fe=l.  (§138) 

«/      b<,  a-       b^  ^  ■' 


Til  10    HYl'KlUiOl-A.  179 

169.  The  equatloti  of  tlie  polar  of  the  pole  (Ji,  k)   icltk 

regard  to  the  hijperbola  is 

lix      kii       .  ^ .  ^ ,,, 

— -7T  =  1.  (§139) 

a^       O'  ^  ^ 

The  tangent  and  chord  of  contact  are  particular  cases  of 

the  polar,  and  the  i)ruposition  of  §  74  holds  true  for  poles 

and  polars  with  regard  to  the  hyberbola. 

170.  The  equation  of  a  diameter  of  ait  hupc.rhula  is 

y  =  ,;?;'.  (S"i) 

in  which  in  is  the  slope  of  its  chords. 

171.  If  m'  is  the  slope  of  the  diameter  bisecting  the 
chords  parallel  to  the  diameter  //  =  nix,  then  (§  142) 

iiini'  —  -^,'  r4ol 

a-  >-     -■ 

Since  ??i  and  m'  are  alike  involved  in  [45],  it  follows  that 

If  0716  diameter  bisects  all  chords  par allel  to  another,  the 

second  diameter  will  bisect  all  chords  parallel  to  the  first. 

Two  such  diameters  are  called  Conjugate  Diameters. 

172.  From  [45],  the  slopes  of  two  conjugate  diameters 
must  agree  in  sign  ;  hence, 

Ttvo  conjxKjate  diameters  of  an  hyperbola  lie  on  the  same 
side  of  the  coiijagate  axis,  and  their  included  angle  is  acute. 

Also,  if  ni  in  absolute  magnitude  is  less  than  -,  then  in' 

b  " 

must  be  greater  than--     But  the  slope  of  the  asymptotes  is 

b  ^^ 

equal  to  ±  —     Therefore, 

Two  conjugate  diameters  lie  on  opposite  sides  of  the  asymp- 
tote in  the  same  quadrant :  and  of  two  conjugate  diameters, 
one  meets  the  curve  in  real  points  and  the  other  in  imaginary 
jtoints  (§  159). 


180  ANALYTIC    GEOMETRY. 

173.  The  length  of  a  diameter  that  meets  the  hyperbola 
in  real  points  is  the  length  of  the  chord  between  these  points. 

If  a  diameter  meets  the  hyperbola  in  imaginary  points, 
that  is,  does  not  meet  it  at  all,  it  will  meet  tlie  conjugate 
hyperbola  iu  real  points  (§  159);  and  its  length  is  the  length 
of  the  chord  between  these  points.  But  from  §  159  we 
know  that  if  a  diameter  meet  one  of  the  hyperbolas  in  the 
imaginary  point  (AV—  1,  k^J—V),  it  will  meet  the  other  in 
the  real  point  (Ji,  k);  hence,  the  length  of  the  semi-diameter, 
which  is  V/r  +  k^,  is  known  from  the  imaginary  coordinates 
of  intersection. 

174.  The  equations  of  an  hyperbola  and  its  conjugate 
differ  only  in  the  signs  of  a^  and  b^.  But  this  interchange 
of  signs  does  not  affect  the  equation 

,      b' 

mm  =  — • 

Therefore,  if  tivo  diameters  are  conjugate  with  respect  to 
one  of  two  conjugate  hyperbolas,  they  ivill  be  conjugate  with 
respect  to  the  other. 

Thus,  let  POP'  and  QOQ'  (Fig.  71)  be  two  conjugate 
diameters.  Then  POP'  bisects  all  chords  parallel  to  QOQ' 
that  lie  icithin  the  branches  of  the  original  hyperbola  and 
betioeen  tlie  branches  of  the  conjugate  hyperbola;  and  QOQ' 
bisects  all  chords  parallel  to  POP'  that  lie  within  the 
branches  of  the  conjugate  liyperbola  and  between  the 
branches  of  the  original  hyperbola. 

From  the  above  theorem  it  follows  immediately  that 

If  a  straight  line  meets  each  of  two  conjugate  hyperbolas  in 
two  real  jioints,  the  two  jjortlons  of  the  line  contained  between 
the  hyperbolas  are  equal  (thus,  BD^=  B'D',  Fig.  71). 

175.  The  tangent  di'atvn  through  the  end  of  a  diameter  is 
parallel  to  the  conjugate  diameter  (§  143). 


THE    HYPERBOLA. 


181 


176.    Having  given  the  end  (^j,  ?/i)  of  a  dianiete?-,  to  find 
the  end,  (./•.,,  y.,)  of  the  conjugate  diameter. 


Fig.  71. 


Let  (xi,  ]/i)  be  on  the  given  hyperbola,  then  (arg,  2/2)  is  on 

the  conjugate.     The  slope  of  the  tangent  at  (xi,  1/1)  is  -7-^; 

hence,  the  equation  of  the  diameter  conjugate  to  the  diame- 
ter through  (xi,  1/1)  is 

y=^x.  (1) 

a-i/i 
Now  (xo,  1/2)  is  on  the  diameter  (1)  and  also  on  the  con- 
jugate hyperbola ;  lience,  we  have 


2/2  = 


1. 


a'l/i  a^       0- 

Solving  equations  (2)  for  Xo  and  7/2,  we  obtain 
a  b 


(2) 


X2  '■ 


l/u 


y.2  =  ±-xy. 
a 


182  ANALYTIC    GEOMETRY. 

The  positive  signs  belong  to  one  end,  and  the  negative 
signs  to  the  otlier  end,  of  the  conjugate  diameter. 

177.  To  find  flic  equation  of  an  liijperlmla  referred  to  any 
initr  of  eoirjiigate  dimneters  as  axes  of  coordinates. 

From  the  symmetry  of  the  curve  with  respect  to  each,  of 
the  new  axes,  tlie  required  equation  must  be  of  the  form 

Denoting  tlie  intercepts  of  the  curve  on  the  new  axes  by 
a'  and  b'  V—  1  (§  172),  we  obtain 

Whence,  ^^--^^l  (1) 

is  tlie  required  equation,  in  which  a'  and  V  are  semi-conju- 
gate diameters. 

Since  the  form  of  equation  (1)  is  the  same  as  that  of  the 
equation  referred  to  the  axes  of  the  curve,  it  follows  that 
all  formulas  that  have  been  obtained  without  assuming 
the  axes  of  coordinates  to  be  at  right  angles  to  each  other 
hold  good  when  the  axes  of  coordinates  are  any  two 
conjugate  diameters.  Por  example,  the  equation  of  the 
asymptotes  of  the  hyperbola  represented  by  equation  (1)  is 

0  9 

and  the  equation  of  the  tangent  is 

178.  The  tangents  through  tlie  ends  of  two  conjugate 
diameters  meet  in  the  asymptotes. 

The  equations  of  these  tangents  referred  to  the  conjugate 
diameters  are 

a;  =  ±  «',      y  =  ±b'. 


THE    HYPKKBOLA.  183 

Hence,  tlieir  intersections  are  («',  b'),  (a',  —  b'),  ( —  a',  b'), 
and  ( — a',  — b').  Hut  these  points  evidently  lie  upon  the 
asymptotes,  or  the  locus  of  (2)  in  §  177. 

179.  //'  6  denotes  tlie  aiKjle  formed  by  two  covj agate  semi- 
diametei's,  and  a'  and  b'  their  lengtlis,  then  sin  ^  =  -y-,- 

Substituting  ^V  —  1  for  b,  and  i'V  —  1  for  ^' in  equation 
(1)  of  §  147  and  cancelling  the  imaginary  factorS;  we  obtain 
the  above  result. 

CoR.  1.  Since  Irt'i^'sin  ^  =  4«^,  the  parallelogram  SES'R' 
(Fig.  71)  equals  the  rectangle  on  the  axes  of  the  curve. 

CoR.  2.    The  length  of  the  perpendicular  from   0  upon 

the  tangent  SPIi=  OF  sin  OFS  ^a'  sin  e  =  '-^- 

b 

Cor.  3.    From  §§  145,  155,  177,  we  have 
a'-^  —  b'''^a'--b\ 

180.  If  a  straight  line  cats  an  hyperbola  and  its  asymp- 
totes, the  portions  of  the  line  intercepted  between  the  curve  and 
its  asymptotes  are  equal. 

Let  CC  (Fig.  72)  be  the  line  meeting  the  asym})totes  in 
C,  C'and  the  curve  in. 7*,  B',  and  let  the  equation  of  the 
line  be 

y  =  vix-\-c.  (1) 

Let  M  be  the  middle  point  of  the  chord  BB' ;  then 
(§  170)  the  equation  of  the  diameter  through  vi  is 

^  =  ^.-  (2) 

By  combining  equation  (1)  with  the  equations  of  the 
asymptotes,  we  obtain  the  coordinates  of  the  points  C  and 
C ;  taking  the  half-sum  of  these  values,  we  get  for  the  coordi- 
nates of  the  point  halfway  between  C  and  C  the  values 


184 


ANALYTIC    GEOMETRY. 


05  = 


U^  —  m^d^ ' 


y- 


V'c 


U'  —  m^a^ 


These  values  satisfy  equation  (2)  ;  therefore,  the  point 
halfway  between  C  and  C"  coincides  with  M\  therefore, 
MC=  MC.     And  since  MB=MB\  therefore,  BC=  B'C. 


Dy 


Fig.  72. 


Cor.  Let  CC  be  moved  parallel  to  itself  till  it  becomes 
a  tangent  at  P,  meeting  the  asymptotes  in  R,  S;  then  the 
points  B,  B'  coincide  at  F,  and  we  have  PB  =  PS.    Hence, 

The  portion  of  a  tanr/ent  intercepted  by  the  asymptotes  is 
bisected  by  the  p)oint  of  contact. 

181.  The  following  method  of  showing  that  an  hyperbola 
has  asymptotes,  and  finding  their  equations,  is  more  general 
than  the  method  given  in  §§  159,  160. 


THE    HYPERBOLA.  185 

The  abscissas  of  the  points  where  the  straight  line  y  =  mx 
-\-  e  meets  an  hyperbola  are  found  by  solving  the  equation 

x^      (mx  -f  cy 


a^  b'' 


1, 


Now,  from  Algebra  we  know  that  as  the  coefficients  of 
x^  and  X  in  (1)  approach  zero,  both  roots  of  (1)  increase 
without  limit.     Hence,  each  root  becomes  infinity  when 
V^  —  nvii?  =  0,  and  Imc  =  0, 

or  when  m  =  ±  -,  and  c  ^ 0. 

a 

Therefore,   ij^^±i-x  are  asijmptotes  to  the  hyperbola. 

If  only  b-  —  m^a^  =  0,  then  m  ^  ±  -,  the  line  is  parallel  to 

an  asymptote,  and  one  root  of   (1)  is  infinity,  while  the 

,       .        h'  +  c" 

other  IS 

Zmc 

Hence,  a  right  line  parallel  to  an  asymptote  meets  the 
hyperbola  in  only  one  finite  point. 

182.  To  find  the  equation  of  an  hyperbola  referred  to  the 
asymptotes  as  axes  of  coordinates. 

Let  the  lines  OB,  OC  (Fig.  73)  be  the  asymptotes,  A  the 
vertex  of  the  curve,  and  let  the  angle  AOC=^a. 

Let  the  coordinates  of  any  point  P  of  the  curve  be  x,  y 
when  referred  to  the  axes  of  the  curve,  and  x\  y'  when 
referred  to  OB,  OC  as  axes  of  coordinates. 

Draw  PMl_to  OA,  PN\\  to  CO  ;  then 

X  =  ON  cos  a  -\-  NP  cos  a  =  {x'  +  //')  cos  a, 
y  =  NP  sin  a  —  ON  sin  a=  (y'  —  x')  sin  a. 


186 


ANALYTIC    GEOMKTKY. 


Fig.  73. 


Hence,  by  substituting  in  [40],  we  obtain 
(.rJ  -{-  //')-  cos-  a      ill'  —  x'y  sin^tt 


b- 


1. 


But 


AD 


cos  a: 


OD      Vft2  +  i2 
OA  a 


OB       -sJa'-\-b'^ 
Substituting  these  values,  and  dropping  accents,  we  have 
4:XU  =  H^  +  bK  [4G] 

Cor.  1.    The  equation  of  the  conjugate  hyperbola  is 
4xy=i  — (rt-  +  Z--). 

2ab 
CoR.  2.    Sin  COT?  =  sin  2^  =  2  sin  «  cos  a=    ,  ,   ,,• 

a'  +  ¥ 

It  a  =  b,  sin  COB  =  1 ;  therefore,  COB  =  ^ir. 

Cor.  3.    Let  {x^,  y^)  denote  P  (Fig.  72),  referred  to  the 
asymptotes  ;  then 

OS  X  OR  =  20ffX  2 IIP  =  4a-i//i  =  a^  +  b\ 

That  is,  tJie  product  of  the  intercepts  of  a  tangent  on  the 
asymjjtotes  is  equal  to  the  sum  of  the  squares  of  the  semi-axes. 


THK    HYPERBOLA. 


187 


CoK.  4.    In  Fig.  72,  the  area  of  tlie  triangle  ROS  equals 

h  OS  X  OB  sin  ROS  =  h  (a'  +  b')  ^?^..  =  ah. 

That  is,  the  area  of  the  triangle  formed  by  any  tangent 
and  the  aaymptotes  is  equal  to  the  product  of  the  sevii-axes. 

183.    The  polar  of  the  focus  («e,  0)  is 


Fig.  74. 

Hence,  if  OD  is  taken  so  that 

OF:  OA=OA:OD, 

^hen  DN  perpendicular  to  OF  is  the  polar  of  F,  and  is 

called  a  Directrix  of  the  hyperbola.     In  like  manner  we  may 

construct  i>'X',  or  the  directrix  corresponding  to  the  focus  F'. 

Cor.    As  in  §  150  we  may  prove  that 

_PF 


188  ANALYTIC    GEOMETRY. 

Whence,  the  hyperbola  may  be  defined  as 
The  locus  of  a  jJoint  whose  distances  from  a  fixed  point  and 
a  fixed  straight  line  hear  a  constant  ratio  greater  than  unity. 

1 84.    To  find  the  polar  equation  of  an  hyperbola,  the  left- 
hand  focus  being  taken  as  pole. 

If  X  is  reckoned  from  the  centre,  and  we  write 

p=^ex-\-  a,  (1) 

p  will  be  positive  or  negative  according  as  the  point  is  on 
the  right  or  left-hand  branch. 

Now         x^ p cos  6  —  c^=p cos  6  —  ae. 

Whence,  by  substitution  and  reduction, 

e  cos  6  —  1  "-      -■ 

From  (1)  we  know  that  a  point  is  on  the  right  or  left- 
hand  branch,  according  as  p  in  [47]  is  positive  or  negative  ; 

that  is,  according  as  cos  ^  >  or  <!  — 

If  0  =  0,  p  =  ae^a  =  F'0+OA  =  F'A  (Fig.  70). 

If  e  cos  0  —  1  =  0,  or  ^  =  cos~'  -,  p  =y:>,  as  it  should, 

e 

since  in  this  case  the  radius  vector  is  ||  to  the  asymptote. 
If  6  =  ^TT,  p  ^  —  a  (e-  —  1)=  —  semi-latus  rectum. 

If  e  =  ^,     p=      a  —  ae  =  —  F'A'. 

Exercise  39. 

1.  What  is  the  polar  of  the  point  (—9,  7)  with  respect 
to  the  hyperbola  7x-  — 12/  =  112  ? 

2.  Find  the  equations  of  the  directrices  of  an  hyperbola. 

3.  Find  the  angle  formed  by  a  focal  chord  and  the  line 
that  joins  its  pole  to  the  focus. 


THE    HYPERBOLA.  189 

4.  Find  the  pole  of  the  line  Ax-]- B)/-{-  C  =  0  with  re- 
spect to  an  hyperbola. 

5.  Find  the  polar  of  the  right-hand  vertex  of  an  hyper- 
bola with  respect  to  the  conjugate  hyperbola. 

6.  Find  tlie  distance  from  the  centre  of  an  hyperbola 
to  the  point  where  the  directrix  cuts  the  asymptote, 

7.  If  (xi,  iji)  and  (x.2,  ijo)  are  the  ends  of  two  conjugate 
diameters,  then 

X1X2      jh!h  _  ^ 

8.  The  equation  of  a  diameter  in  the  hyperbola  2ox^  — 
16^/^^400  is  3>/  =  x.  Find  the  equation  of  the  conjugate 
diameter. 

9.  In  the  hyperbola  49a'^  —  4y"  =  196,  find  the  equation 
of  that  chord  which  is  bisected  at  the  point  (5,  3). 

10.  Find  the  length  of'the  semi-diameter  conjugate  to 
the  diameter  y  =  3x  in  the  hyperbola  9x-  —  4^^  =  36. 

11.  Two  tangents  to  an  hyperbola  at  right  angles  intersect 
on  the  circle 

a-^  -[-  ?/^  =  «^  —  b^. 

12.  Tangents  at  the  extremities  of  any  chord  of  an  hyper- 
bola intersect  on  the  diameter  which  bisects  that  chord. 

13.  Prove  that  FQ  (Fig.  71)  is  parallel  to  one  asymptote 
and  bisected  by  the  other. 

14.  An  asymptote  is  its  own  conjugate  diameter. 

15.  The  conjugate  diameters  of  an  equilateral  hyperbola 
are  equal. 

16.  Having  given  two  conjugate  diameters  in  length  and 
position,  to  find  by  construction  the  asymptotes  and  the 
axes. 

17.  To  draw  a  tangent  to  an  hyperbola  from  a  given 
point. 


190  ANALYTIC    GKOMKTKY, 

18.  Find  the  equation  of  a  tangent  at  any  point  (x^,  v/i) 
of  the  h^^perbola  Axt/  =  (r-{-  Ir. 

19.  Find  the  equation  of  an  hyperbola,  taking  as  the 
axis  of  ij 

(i)  the  tangent  through  the  left-hand  vertex ; 
(ii)  the  tangent  through  the  right-hand  vertex. 

20.  Find  the  polar  equation  of  an  hyperbola,  taking  the 
right-hand  focus  as  pole. 

21.  Find  the  polar  equation  of  an  hyperbola,  taking  the 
centre  as  pole. 

22.  To  find  the  centre  of  a  given  hyperbola. 

23.  The  distance  from  a  fixed  point  to  a  fixed  straight 
line  is  10.  Find  the  locus  of  a  point  which  moves  so  that 
its  distance  from  the  fixed  point  is  always  twice  its  distance 
from  the  fixed  line. 

24.  Show  that  the  locus  of  x^  —  \if  —  2x—  16y  —  19  =  0 
is  an  hyperbola ;  find  its  centre  and  axes. 

25.  If  A  and  B  have  unlike  signs,  prove  that  the  locus 
of  Ax^  -\-  Bif  +  I)x  +  E[i  +  F^^  0  is  in  general  an  hyperbola 
whose  axes  are  parallel  to  the  coordinate  axes ;  and  deter- 
mine its  semi-axes. 

26.  Through  the  point  (—  4,  7)  a  straight  line  is  drawn 
to  meet  the  axes  of  coordinates,  and  then  revolved  about 
this  point.  Find  the  locus  of  the  point  midway  between 
the  axes. 

27.  A  straight  line  has  its  ends  in  two  fixed  perpen- 
dicular lines,  and  forms  with  them  a  triangle  of  constant 
area  d\     Find  the  locus  of  the  middle  point  of  the  line. 

28.  The  base  «.  of  a  triangle  is  fixed  in  length  and  posi- 
tion, and  the  vertex  so  moves  that  one  of  the  base  angles 
is  always  double  the  other.     Find  the  locus  of  the  vertex. 


CHAPTER  VIII. 
LOCI  OF  THE  SECOND  ORDER. 

185.  The  loci  represented  by  equations  of  the  second 
degree  that  are  not  of  the  first  order  are  called  Loci  of  the 
Second  Order. 

In  the  preceding  chapters  we  have  seen  that  the  circle, 
parabola,  ellipse,  and  hyperbola  are  loci  of  the  second  order. 
We  will  now  inquire  whether  there  are  other  loci  of  the 
second  order  besides  the  four  curves  just  named  ;  in  other 
words,  we  will  determine  what  loci  may  be  represented 
by  equations  of  the  second  degree. 

For  this  purpose  we  shall  write  the  general  equation  of 
the  second  degree  in  the  form 

Ax^  -H  Bir  +  Cri/  +  Z)a;  +  %  +  F=  0,  (1) 

and  shall  assume  that  the  axes  of  coordinates  are  rectan- 
gular. This  assumption  will  in  nowise  diminish  the  gener- 
ality of  our  conclusions ;  for  if  the  axes  were  oblique,  we 
could  refer  the  equation  to  rectangular  axes,  and  this  change 
would  not  alter  the  degree  of  the  equation  or  the  nature  of 
the  locus  which  it  represents  (§  91).  • 

186.  To  find  the  condition  that  the  rjeneral  equation  of 
the  second  degree  may  represent  two  loci  of  the  first  order. 

To  do  this  let  us  solve  (1)  with  respect  to  one  of  the 
variables.     Choosing  y  for  this  purpose,  we  obtain 

Cx  +  E       1      , . 

r         y  = ^^^^^Lx^  +  Mx^N,  (2) 

where  L=C^  —  iAB,  iI/=  2 ( CE  —  2BD) ,  N=^  E-  —  \BF. 


192  ANALYTIC    GKOMKTRY, 

If  Lx^ -\- Mx -\- N  is  a  perfect  square,  then  tlie  locus  of 
(2),  or  (1),  will  be  two  loci  of  the  first  order. 

Now,  from  Algebra,  we  kuow  that  the  condition  tliat 
Lx^  +  3fx  +  N  should  be  a  perfect  square  is 

or,  substituting  the  values  of  L,  M,  and  N,  we  have 

(CE-2BDy-{C^  —  4.AB)  (E'-4:BF)  =  0, 
or  F(C^  —  4:AB)-\-AE^  +  BD^—CI)i;=0.  (3) 

The  quantity  on  the  left-hand  side  of  equation  (3)  is 
usually  denoted  by  A,  and  is  called  the  Discriminant  of 
equation  (1). 

This  same  result  was  obtained  by  a  more  general  method 
in  §  57  ;  hence, 

Whenever  A  =  0,  eqiiation  (1)  represents  two  loci  of  the 
first  order.  These  loci  may  be  readily  determined  by 
resolving  (1)  into  two  simple  equations  in  x  and  y. 

CENTRAL  CURVES.  2  NOT  ZERO. 

187.  A  centre  of  a  curve  is  a  point  that  bisects  every 
chord  passing  through  it.  Loci  are  classified  as  Central 
and  Non-Central,  according  as  they  have  or  have  not  a 
definite  centre.  The  circle,  ellipse,  and  hyberbola  belong 
to  the  first  class,  the  parabola  to  the  second. 

188.  To  find  the  equation  of  the  central  loci  represented, 
by  eq^iation  (1)  referred  to  their  centre. 

To  do  this  let  us  change  the  origin  to  the  point  {h,  k), 
and  then  so  choose  the  values  of  h  and  k  that  the  terms 
involving  the  first  powers  of  x  and  y  will  vanish.  Making 
the  change  by  substituting  in  (1)  x-\-h  for  x,  and  y-\-k  for 
y,  we  find  that  the  coefficients  A,  B,  and  C  remain  unaltered, 
and  we  may  write  the  transformed  equation 

Ax^  +  By""  +  Cxy  +  D'x  +  E'y  =  R,  (4) 


LOCI    OF    THE    SECOND    ORDER.  193 

wh  ere  D'  =  2  A  h  +  Ck  +  D, 

E'  =  2Bk+Ch-\-E, 

R=—  [A/i"  +  Bk-  +  Ckk  +  Dh  +  m-  +  F^. 
The  values  of  h  and  k  that  will  make  1)'  and  E'  vanish 
are  evidently  found  by  solving  the  equations 
2Ah+Ck-{-D  =  0, 
2Bk-\-Ch-\-E=0, 

CE-2BB      j_CD  —  2AE 
and  are        ^'^-4^^_C2'     ''-AAB-C' 

If  4:AB  —  C^,  denoted  by  2,  is  not  zero,  these  values  of  h 

and  k  are  finite  and  single,  and  equation  (4)  may  be  writttui 

Ax^  +  Bi/^-\-Cxi/  =  E.  (5) 

From  the  form  of  (5)  we  see  that  if  (x,  y)  is  a  point  in 
its  locus,  so  also  is  ( — x,  — //) ;  that  is,  the  new  origin  (Ji,  k) 
is  the  centre  of  the  locus.     Hence, 

Whe7i  2  is  not  zero,  equation  (1)  can  he  reduced  to  the 
form  of  (5),  and  represents  central  curves. 

When,  however,  2  =  0,  the  values  of  h  and  k  become 
infinite  or  indeterminate,  and  the  locus  of  (1)  lias  no  defi- 
nite centre.     Hence, 

When  2  =  0,  (1)  cannot  be  reduced  to  the  form  of  (5),  and 
rejiresents  no7i-central  curves. 

The  value  of  R  can  be  reduced  to  the  following  useful 
form,  which  shows  also  that  R  and  A  vanish  together. 
R  =  —  [Ah'  +  Bk-  +  Ch  k  +  l)h  +  Ek  +  F^ 

=  -i[(2Ah+Ck  +  D)h 

+  (2Bk  +  Ch  -\-  E)k  +  Dh  +  Ek  +  2/^] 

=  —  i  {D'h  +  Ek  +  Dh  +  Ek  +  2F) 

=  -^(I)h-\-Ek-\-2F) 

_        2BD'—CI)E-\-2AE?—CnE-\-2F(C'-AAB) 

~     *  C'  —  4:AB 

_      A 

~      2' 


194  ANALYTIC    GKOMETRY. 

189.  To  reduce  (5)  to  a  known  form  hij  rausinf/  the  tern„ 
in  xy  to  disappear. 

For  this  purpose  we  change  the  direction  of  the  axes 
through  an  angle  0,  keeping  the  origin  unaltered,  and  then 
determine  the  value  of  0  by  putting  the  new  term  that 
involves  xy  equal  to  zero. 

The  change   is   made  by  substituting  for  x  and  y,   in 
equation  (5),  the  respective  values  (§  86), 
X  cos  6  —  y  sin  6, 
X  sin  6-]r  y  cos  6  ; 
and  equation  (p)  now  beco]nes 

Px''-^ny-+C'.ry  =  B, 
where  P  =  A  cos-  6  -f-  J^  sin^  0-}-  C  sin  0  cos  6,         (&'^ 

Q=^  A  sin-  6-\-  B  cos-  d  —  C  sin  6  cos  6,         (7) 
C'  =  2{B  —  A)  sin  9  cos  0  +  C(cos-  $  —  sin^  6).   (8) 
Putting  C'=^0,  we  obtain,  by  Trigonometry, 

(A  —  B)  sin  26  —  C  cos  20  =  0,  (9) 

or  tan  2^  =  —^-.  (10) 

A  —  x> 

Since  any  real  number,  positive  or  negative,  is  the  tan- 
gent of  some  angle  between  zero  and  tt,  equation  (10)  is 
satisfied  by  some  value  of  9  between  zero  and  •^tt.  In  what 
follows  we  shall  use  the  simplest  root  of  (10). 

I>y  this  transformation,  equation  (5)  is  reduced  to  the 
form  p^.2^^y._^^  (jl) 

of  which  the  discussion  will  be  found  in  the  next  section. 
Cor.  1.    The  values  of  P  and  ()  in  terms  of  A,  B,  C  may 
be  found  as  follows  : 

From  (G)  and  (7),  by  addition  and  subtraction, 

P^Q  =  A  +  B,  (12) 

P  —  ^  =  (^  —  J?)  cos  2^  +  C  sin  26.  (13) 


LOCI    OF    THE    .SECOND    ORDER.  195 

Equation  (9)  may  be  written 

Oz=(A  —  B)  sin  26  —  0  cos  26.  (14) 

Adding  the  squares  of  (13)  and  (14),  we  have 

(P-Qf  =  (A-Br+C',  (15) 

or  P-Q^±\/{A-Bf+C'i  (IG) 

Whence,  from  (12)  and  (16), 

P  =  ^[A  +  B±  V(^ -  By  +  C%  (17) 

Q  =  i[.l  +  P  rp  V(.4  -  By  +  C'l  (18) 

These  values  of  F  and  Q  are  evidently  always  real. 
Cor.  2.    By  squaring  (12)  and  subtracting  (15),  we  obtain 

4FQ  =  4.AB  —  C  =  X  (19) 

Hence,  I*  and  Q  have  like  or  unlike  sir/tis,  accordintj  as  % 
is  positive  or  negative. 

Cor.  3.  In  applying  formvdas  (17)  and  (18),  the  question 
arises  which  sign  before  the  radical  should  be  used.  If  in 
(13)  we  substitute  for  cos  26,  its  value  obtained  from  (14), 
we  have 

p      ^^[(^-/?)^+C'^]sin2g 
C/ 

Since  the  numerator  of  the  fraction  is  always  positive, 
P — Q  must  have  the  same  sign  as  6';  that  is,  the  upper 
or  lower  sign  in  (16)  must  be  taken  according  as  C  is  posi- 
tive or  negative. 

Hence,  the  upper  or  lower  signs  in  (17)  and  (18)  are  to  he 
taken  accordinr/  as  C  is  positive  or  negative. 

190.  The  nature  of  the  locus  represented  by  equation 
(11)  depends  upon  the  signs  of  /*,  Q,  and  R.  There  are 
two  groups  of  cases,  according  as  2  is  positive  or  negative, 
and  three  cases  in  each  group. 


196  ANALYTIC    GEOMETRY. 

Group  1.     2  Positive. 
In  this  group,  P  and  Q  must,  by  (19),  agree  in  sign. 
Case  1.    If  R  agrees  in  sign  with  P  and  Q,  then,  by  §  124. 

the  locus  is  an  ellipse  whose  semi-axes  are  a/—  and  \-7^- 

li  P=  Q,  the  locus  is  a  circle. 

Case  2.  If  R  differs  from  P  and  Q  in  sign,  no  real  values 
of  X  and  ij  will  satisfy  (11),  so  that  no  real  locus  exists. 

Case  3.    If  2t  =  0,  the  locus  is  the  single  point  (0,  0). 

Group  2.     %  Negative. 

In  this  group,  P  and  Q,  by  (19),  must  have  unlike  signs. 

Case  1.  If  ^agrees  in  sign  with  P,  we  may, by  division 
(and  by  changing  the  signs  of  all  the  terms  if  necessary), 
put  equation  (11)  into  the  form  of  equation  [40],  page  171. 
Therefore,  the  locus  is  an  hyperbola,  with  its  transverse, 
axis  on  the  axis  of  x,  and  havinar  for  semi-axes 


«=V] 


Case  2.  If  R  agrees  in  sign  with  Q,  we  may,  by  divisioi. 
(and  by  change  of  signs  if  necessary),  put  equation  (11) 
into  the  form  of  equation  (1),  page  172.  Therefore,  the  locus 
is  an  hyperbola,  with  its  transverse  axis  on  the  axis  of  y. 

Case  3.  If  i?  =  0,  the  locus  consists  of  tioo  straight  lines, 
intersecting  at  the  origin,  and  having  for  their  equations 


NON-CENTRAL    CURVES.     2  =  0. 

191.    To  determine  the  locus  of  (1)  ivhen  A  and  2  are  loth 
zero. 


LOCI    OF    THE    SECOND    ORDER.  197 

If  A  =  2  =^  0,  then  from  the  first  form  of  A  in  §  186  we 
must  have          ^j,j  -2BD  =  0.  (20) 

Hence,  L  =  3I=0,  (§186) 

and  (2)  becomes 

2Bi/  +Cxi-JSzf  '\lE^  —  4.BF=  0,  (21) 

which  represents  two  parallel  straight  lines,  two  coincident 
straight  lines,  or  no  locus,  according  as  E^  —  4:BF';>,  =, 
or<0. 

When  4:AB —  C^  =  0,  if  C  is  not  zero,  neither  A  nor  B 
can  be  zero;  if  C  =  0,  either  A  or  B  must  be  zero,  but  both 
cannot  be,  since  if  A=^B  =  C  =  0,  (1)  is  no  longer  of  the 
second  degree. 

When  C=^B=0,  by  solving  (1)  for  x,  and  introducing 
the  above  condition,  we  should  obtain,  instead  of  (21)  its 
corresponding  equation. 


2 Ax  -\-Ci/-\-D:f  \/l)^—4:AF=  0,  (22) 

whose  locus  is  also  two  parallel  straight  lines.     Hence, 

When  A  and  2  are  both  zero,  equation  (1)  represents  two 
parallel  straight  lines,  two  coincident  straight  lines,  or  no  locus. 

Cor.  Eliminating  B  between  2  =  0  and  (20),  we  obtain 
CD  —  2AE=0.  (23) 

In  like  manner  (20)  follows  from  2  =  0  and  (23). 
From  these  results,  and  the  values  of  h  and  k,  we  learn  that 
(i)    When  A=2=0,  h  and  k  are  both  indeterminate, 
and  conversely. 

(ii)    The  values  of  h  and  k  are  indeterminate  together. 

192.  To  determine  the  locus  of  (1)  ivhe7i  2  is  zero  and  A 
is  not  zero. 

We  simplify  (1)  by  first  making  the  term  in  ary  disappear 
by  proceeding  exactly  as  in  §  189;  that  is,  by  turning  the  axes 


19<S  ANALYTIC    GEOMETRY. 

through  an  angle  9,  the  value  of  which  is  determined  by  the 
equation  f-i 

^^^^^  =  jZZb-  (24) 

If  F,  Q,  U,  Frei)resent  the  new  coefficients  of  x^,  if,  x,  y, 
respectively,  P  and  Q  will  have  values  identical  with  those 
of  P  and  Q  given  in  §  189,  and 

U=      JD  cos  e  +  i:  sin  9,  (25) 

V=  —  D  sin  9  +  1!:  cos  9.  (26) 

Since  C-=4:AB,  from  (17)  and  (18)  we  have,  when  C  is 
positive,  p_^^^   g_0. 

When  C  is  negative, 

F  =  0,  Q  =  A-\-B, 
and  (1)  assumes  the  form 

Qf  +  Ux  +Vy  +  F=0.  (27) 

To  further  simplify,  we  divide  by  Q,  and  obtain 
U         V        F 
^^+^^-  +  ^^+^  =  '' 

If  we  now  take  as  a  new  origin  the  point 
4:QF-V^         V 


^UQ  2Q 

equation  (28)  becomes 

U 

which  represents  a  parabola  whose  axis  coincides  with  the 
axis  of  X,  and  which  is  situated  on  the  positive  or  the  negative 
side  of  the  new  origin,  according  as  U  and  Q  are  unlike  or 
like  in  sign  (§  94). 


liOCI    OF    THE    SECOND    OIIDEK.  199 

The  vertex  of  the  parahuhi  is  the  new  origin,  and  the 
parameter  is  equal  to  the  coeiiicient  of  x  in  the  equation 
of  the  curve. 

This  hist  transformation  is  possible  except  when  fJ^=0  ; 
but,  when  U=^  0,  (27)  evidently  represents  two  parallel 
straight  lines. 

Suppose  that  C  is  positive.  Then  the  general  equation 
becomes 

Fx^-i-  Ux-i-V>/+F=0.  (29) 

And  this,  by  changing  the  origin  to  the  point 

4:PF—  U^  U 


(■ 


4.VP  IP 


becomes  cc^  =  —  ^  y. 

This  represents  a  parabola  having  the  axis  of  y  for  its 
axis,  and  placed  on  the  j/oslf ire  or  the  neyafire  side  of  the 
new  origin,  according  as  V  and  P  are  unlike  or  like  in  sign. 

It  should  be  noted  that  the  value  of  P  or  Q,  when  not 
zero,  is  A  +  P-  The  values  of  U  and  V*  can  be  found  from 
(25)  and  (26). 

*  We  may  obtain  the  values  of  U  and  T"  iu  terms  of  the  origmal 
coefficients,  as  follo%YS : 

From  (24)  we  find,  by  Trigonometry, 

-(A-  B)  ±  -J  (A  -  BY-  +  C2 

tan  6  = jf •.• 

Introducing  the  condition  iAB  —  C^,  we  obtain 

2A 
tan  6  = —5     if  C  is  negative  ; 

2B  ..  ^  . 

=  -^1  if  C  IS  positive  ; 

whence,  if  C  is  negative, 

2^  -C 

sin  e 


V442  -I-  (>2  V4^2  +  02 

And  if  C  is  positive, 


200 


ANALYTIC    GEOMETRY. 


If  C^A  =  0,  the  given  equation  is  of  the  form  of  (27), 
its  locus  is  a  parabola  and  can  be  found  as  that  of  (27).  If 
6'  =  ^  =  0,  the  given  equation  is  of  the  form  of  (29),  and 
its  locus  is  a  parabola. 

193.  The  main  results  of  the  investigation  are  given  in 
the  following  Table  : 


Loci 

Represented  by  the  General  Equation  of  the 
Second  Degree. 

Ax^  +  By^-+  Cxy  +  Dx  +  i:y+  F=0. 

CLASS. 

CONDITIONS. 

LOCL'S. 

I. 

Loci 

having  a 

centre. 

S  positive,    A  not  zero. 
S  positive,    A  =  0. 
2  negative,  A  not  zero. 
S  negative,  A  =  0. 

Ellipse,  or  no  locus. 

Point. 

Hyperbola. 

Two  intersecting  straight  lines. 

II. 

Loci  not 

having  a 

centre. 

2  =  0,  A  not  zero. 
S  =  0,  A  =  0. 

Parabola. 

r  Two  parallel  straight  lines, 
]  One  straight  line, 
V  Or  no  locus. 

Thus  it  appears  that  there  are  no  loci  of  the  second  order 
besides  those  whose  properties  have  been  studied  in  the 
preceding  chapters. 


sm  6  = 


2B 


cos  e  = 


V452  +  C2 

By  substitution,  we  obtain  from  (25)  and  (26) 
if  C  is  negative. 


V4B2  +  C2 


if  C  is  positive, 


U  = 


U  = 


2AE  -  CD 
2  BE  +  CD 

^JIiF+~c^ 


v  = 


v  = 


2AD+  CE 

CE-2BD 
■\J4B^  +  C^ 


(30) 
(31) 


•  LOCI    OF    THE    SECOND    ORDER.  201 

194.    Examples.     1.    Determine  t?ie  nature  of  the  locus 
ox^-\-57/-{-2xi/  —  12x  —  12ij  =  0,  (1) 

transform  the  equation  and  construct  it. 

Here        A  =  5,  B=o,  C=2,  D=-12,  E=-12,F=0. 

Whence,  2  =  96,  A  ==1152. 

Hence,  the  equation  represents  an  ellipse  or  no  real  locus. 

Therefore,  the  equation  of  the  locus  referred  to  new 
parallel  axes  through  the  centre  (1, 1)  is  (§  188) 

5x'-  +  2xi/  +  5f-  =  12.  (2) 

To  cause  the  term  in  x)/  to  disappear,  we  have 

Whence,  2$  =  90°,     or  ^  =  45°. 


(We  use  the  upper  signs  m  the  values  of  P  and  Q  by  §  189.) 
Hence,  by  §  189  the  equation  of  the  ellipse  referred  to 
its  own  axes  is 

60.2  +  4^^-12,     or  1  +  1  =  1.  (3) 

Whence,  a  =  VS,  b  =  V2,  and  a  lies  on  the  axis  of  i/. 

To  construct  the  equation,  draw  the  axes  OiXi,OiYi  (Fig. 
75);  locate  the  centre  (1, 1).  Through  this  point  O2  draw 
the  second  set  of  axes,  O^X^,  0^  Y„.  Through  O2  draw  the 
third  set  of  axes  OaA'g,  02^3,  making  X0O0X3  equal  to  45°. 

Lay  off  O2A'  =  O.A  =  Vs,  and  0^0^  =  O^B  =  V2.  The 
ellipse  having  BO^  and  A  A'  as  axes  will  be  the  required  locus. 


202 


ANALYTIC    GEOMETRY. 


The  equation  of  the  locus  referred  to  the  axes  OiX^,  Oi  Yi 
is  (1);  its  equation  referred  to  O^Xo,  O^Y^  is  (2);  and  its 
equation  referred  to  O^X^,  O^Y^  is  (3).  In  constructing 
the  locus  it  is  not  necessary  to  draw  the  second  set  of  axes 
C/a JCj,  (/a jfa* 


(1) 


Fig.  75.' 

2.    Determine  the  nature  of  the  locus 

x^-\-?f-  5x1/  +  Sx-  -  20^  +  15=0, 
transform  the  equation  and  construct  it. 
Here  2  =  -21,     A  =  —  21. 

Tlierefore,  the  locus  is  an  hyperbola. 

i?  =  A^2  =  l,     A  =  -4,     k  =  0. 
Hence,  the  first  transformed  equation  is 

x^-\-y"  —  5xi/  =  l. 
In  the  second  transformation, 

^-45°,     P  =  -|,     Q  =  i. 
Hence,  the  equation  of  the  curve  referred  to  its  own  axes  is 

3x'  — 7/  =  — 2, 

from  which  we  see  that  a  =  Vf ,  b  =  V§,  and  a,  or  the  trans- 
verse axis,  lies  on  the  axis  of  y. 


(2) 


LOCI    OF    THE    SECOND    ORDER. 


203 


To  construct  the  equation,  draw  the  axes  OiXi,  OjFi, 
locate  the  centre  ( — 4,  0),  and  through  it  draw  the  third  set 


Fig.  76. 

of  axes  O2X3,  O2YS,  making  A'lOaA's^  45°.  Then  lay  off 
02A'=02A=\/^,  and  O^B  =  O^B' =  ^ ^ ;  and  draw  the 
hyperbola  having  AA'  and  BB'  as  its  transverse  and  conju- 
gate axes  respectively. 

3.  Determine  the  nature  of  tlie  locus 

a;^  +  2.ry-y2-f-8.r +  4^-8  =  0,  (1) 

transform  the  equation  and  construct  it. 

Here  2  =  —  8,  A= — 176;  hence,  the  locus  is  an  hyperbola. 

^  =  22,  h  =  —  3,  /.•  =  —  !; 
hence,  the  first  transformed  equation  is 

x'  -\-  2x7/  —  if-  =  22.  _  (2) 

0  =22|-°,  P=\l2,  Q  =  —  \/2; 
hence,  the  equation  of  the  curve  referred  to  its  own  axes  is 

V2a;2— V2//  =  22.  (3) 

The  hyperbola  is  equilateral,  and  its  transverse  axis  lies 
on  the  axis  of  x.     (Let  the  reader  construct  it.) 

4.  Determine  the  nature  of  the  locus 

x^-\-,f  —  2x!i-\-2x  —  i/  —  l=0,  (1) 

transform  the  equation  and  construct  it. 


204 


ANALYTIC    GEOMETRY. 


Here  2  =  0,  A  is  not  zero. 
Therefore,  the  locus  is  a  parabola. 

0  =  4:5°,  F  =  0,  Q  =  2,  U=^-\l2,  V=  —  %\l2- 

hence,  by  revolving  the  axes  through  an  angle  of  45°,  the 
equation  becomes 

2^"  +  i  V2a;  —  %-^2ij  -1  =  0, 

or  t/2-fV2^+(tV2)^=-iV2^  +  M. 

or  (2^_3V2)2  =  -iV2(x-f|V2).  _         (2) 

Passing  to  parallel  axes  whose  origin  is  (f|V2,  |V2), 
(2)  becomes 

y^  =  -iV2^,  ■  (3) 

the  locus  of  which  is  a  parabola  whose  latus  rectum,  or 
parameter,  is  ^\l2. 


Fig.  77 

To  construct  the  equation,  draw  the  original  axes  O^X^, 
OxYi,  then  draw  the  second  set  of  axes  O1X2,  O1Y2,  making 

XiC>iXj  =  45°. 
Locate  the  new  origin  O3,   (f|V2,  |V2),  and  through  it 
draw  the  third  set  of  axes  O^Xz,  O3  Pg,  to  which  (3)  refers 
the  locus,  which  is  now  easily  drawn. 

5.    Determine  the  nature  of  the  locus 

x^  —  ixy  +  iy^  —  6x-\-  12y  =■  0. 


LOCI    OF    THE    SECOND    ORDER. 


205 


Here  2  =  0  and  A  =  0. 

Factoririg  the  first  member  of  the  equation,  or  solving 
the  equation  for  x  or  y,  we  obtain  as  the  equations  of  these 
lines 

x  —  2ij  =  0,  x  —  2y  —  Q  =  0. 

Hence,  the  locus  is  two  parallel  straight  lines. 
6.    Determine  the  nature  of  the  locus  of 
if'  —  xy  —  Q>x-  —  3.x  +  ?/  =  0. 

Here  2  is  negative,  and  A  =  0. 

Hence,  the  locus  is  two  intersecting  straight  lines.  Their 
equations  are 

y  +  2a;  + 1  =  0,  and  ?/  —  3x  ==  0. 

195.  A  Conic  is  the  locus  of  a  point  whose  distance  from 
a  fixed  point  bears  a  constant  ratio  to  its  distance  from  a 
fixed  straight  line. 

196.  To  find  the  equation  of  a  conic. 

Y 

N 


Fig.  78. 


Let  F  be  the  fixed  point,  and  YY'  the  fixed  straight  line. 
Through  jPdraw  XO  perpendicular  to  YY,  and  use  OA'and 
OFas  axes  of  reference.    Let^:>  denote  the  distance  OF,  and 


20G  ANALYTIC    GEOMETRY. 

e  tlie  constant  ratio ;  then  FP  -=r  NP  =  e,  P  being  any  point 
(a-,  I/)  in  the  curve. 

Now  FP^  =  Fir'  +  MP\ 

But  FP  =  eX  NP  =  ex, 

FM=x—p,  MP  =  ij. 

Therefore,  e^x'^  =  (x  — j'Y  +  if^ 
or  (  1  -  e^)  a:^  +  /  -  2px  +  /^^  =  Q,  (1) 

which  is  the  equation  required. 

CoR.    In  equation  (1),  which  is  of  the  second  degree, 
2  =  4  (1  —  tj-),  and  A  =  4/vV. 

Hence,  when  the  fixed  point  is  without  tlie  fixed  line,  A 
is  not  zero,  and 

If  e  <C  1,  2  >  0,  and  the  conic  is  an  ellipse. 

If  e  =  1,  2  =  0,  and  the  conic  is  a  parabola. 

If  e  >•  1,  2  <C0,  and  the  conic  is  an  hyperbola. 

When  the  fixed  point  is  in  the  fixed  line,  A  ^  0,  and 

If  e  <  1,  2  >  0,  and  the  conic  is  the  point  (0,  0). 

If  e  =  1,  2  =  0,  and  the  conic  is  a  right  line. 

If  « >  1,  2  <  0,  and  the  conic  is  two  intersecting  right 
lines. 

If  e^O,  by  §  61,  the  conic  is  a  cii'cle  or  a  point,  accord- 
ing as  p  is  not  or  is  zero. 

From  §§  92,  150,  183,  it  follows  that  the  fixed  point  is  a 
Focus,  the  fixed  right  line  a  Directrix,  and  the  constant 
ratio  the  Eccentricity,  of  the  conic. 

Exercise  40. 

Determine  the  nature  of  the  following  loci,  transform 
their  equations,  and  construct  them  : 

1.  ^x'-{-2if—2x-\-y-\=0. 

2.  3a;2-f-2j7/  +  3/  — ir.//  +  23  =  0. 


LOCI    OF    TlIK    SKCOND    ORDER.  207 

3.  X-  —  I O.v.v  +  //"  +  •'•  +  //  +  1  =  0. 

4.  x^  -f-  xi/  -{-  //'^  -\-  X  -{-  1/  —  5  =  0. 

5.  if  —  X-  —  y  =  0. 

6.  l+2::c  + 3^2  =  0. 

7.  y-~2xi/-\-x^  —  Sx-\-16  =  0. 

8.  ar  —  2x1/  -\- 1/'^  —  Gx  —  6^  +  9  =  0. 

9.  7/_2x  — 8//  +  10  =  0. 

10.  4a;2  +  V  +  8.r  +  36//  +  4  =  0. 

11.  52cc=^  +  72x//  +  73/  =  0. 

12.  9/  —  4x2  _  8.^  _|_  18^  +  41  =  0. 

13.  y^  —  xy  —  5x-\- 5i/  =  0. 


CHAPTER  IX. 
HIGHER  PLANE   CURVES. 

197.  An  Algebraic  Curve  is  one  whose  rectilinear  equa- 
tion contains  only  algebraic  functions.  A  Transcendental 
Curve  is  one  whose  rectilinear  equation  involves  other  than 
algebraic  functions.  Thus,  the  loci  of  7j=^a^,  ?/  =  tan  x, 
y  =  {a  —  X)  tan  {jttx  -i-  a),  y  =  sin  ~^x  are  transcendental 
curves.  Transcendental  curves  and  all  algebraic  curves 
above  the  second  order  are  called  Higher  Plane  Curves. 

Let  the  symbol  F  (x,  y)  denote  any  rational  integral  func- 
tion of  x  and  y,  of  the  third  or  higher  degree.  If  F  (x,  y) 
breaks  up  into  simple  or  quadratic  factors  in  x  and  y,  the 
locus  of  F  (x,  y)  =  0  consists  of  lines  of  the  first  or  second 
order.  If  F  (x,  y)  does  not  break  up  into  rational  factors  in  x 
and  y,  the  locus  of  F(x,  ?/)  ^=  0  is  a  higher  plane  curve  whose 
order  is  of  the  degree  of  the  equation.     Thus,  the  locus  of 

y^  —  x^^{y  —  x)  (y^  +  a-y  +  x^)  =  0, 
consists  of  the  right  line  y  —  x  =  0  and  the  ellipse  y-  -\-  xy 
-\-  x^  =  0  ;  the  locus  of 

y*-\-xy-{-2y^—2x*—5x"—3=(y'-\-2x^+S)(y^—x^-l)=0, 
consists  of  the  ellipse  y-  -\-  2x^  +  3^0,  and  the  hyperbola 
y^  —  x^  —  1=0;  while  the  locus  of  y^  —  ax -\-  x'^  ^  0  is  a 
higher  plane  curve  of  the  third  order. 

In  this  chapter  we  shall  consider  a  few  of  the  higher 
plane  curves,  some  of  which  possess  historic  interest  from 
the  labor  bestowed  on  them  by  the  ancient  mathematicians. 

198.  The  Cissoid  of  Diodes.  Let  X^(rig.  79)  be  a 
tangent  to  the   circle  XSO  at  the  vertex  of  any  diameter 


HIGHEK    PLANE    CURVES. 


209 


OX ;  let  OR  be  any  right  line  from  0  to  XH  cutting  the 
circle  at  S,  and  take  0F  =  ES;  then  the  locus  of  P,  as  Oli 
revolves  about  0,  is  the  Cissoid. 


To  find  its  equation  referred  to  the  rectangular  axes  OX 
and  OY,  let  OM=x,  MP=ij,  and  OC=CX=  CD  =  r. 

Now,  MP  :  OM ::  NS :  ON.  (1) 

Since  OP  =  ES,   OM=NX. 

Hence,         0N=  OX—  NX=  OX—  0M=  2r  —  x, 
and  NS=  -^ONXNX^  \l{2i — x)x. 

Substituting  these  values  in  (1),  we  obtain 

£C3 


2/^  V(p; 


iC)X 


which  is  the  equation  sought. 


or  y^ 


2r-ac' 


[48] 


210  ANALYTIC    GEOMETRY, 

A  discussion  of  [48]  leads  to  the  following  conclusions: 
(i)    The  curve  lies  between  the  lines  x  =  0  and  x  =  2r. 

(ii)    It  is  symmetrical  with  respect  to  the  axis  of  x. 

(iii)  It  passes  through  the  extremities  of  the  diameter 
perpendicular  to  OX. 

(iv)  It  has  two  infinite  branches  to  which  x  =  '2r  is  an 
asymptote. 

CoR.  1.  To  find  the  polar  equation,  let  6  =  X0F  and 
P=OF; 

^      OX       OS 
then  '^'^=0e'''0X' 

Hence,  p=  OF  =  SE=  OE- 0S=-^  —  2r  cos  6 

^  cos  0 

sin'^O 

cos  9 

Therefore,   p  ^  2r  sin  6  tan  6, 
which  is  the  equation  sought. 

The  cissoid  was  invented  by  Diodes,  a  Greek  mathema- 
tician of  the  second  century  b.c,  for  the  solution  of  the 
problem  of  findlvr/  tivo  mean  proportionals,  of  which  the 
dupUcation  of  the  cube  is  a  particular  case. 

Cor.  2.  To  duplicate  the  cube,  in  Fig.  79,  take  CB  =  2r, 
and  draw  5X cutting  the  cissoid  in  ^;  then,  smceCX=^CB, 
EX=^EQ.    But  from  tlie  o(] nation  of  the  cissoid,  we  have 

W  ==  ^'  =  r^;  therefore,  W  =  20E'. 
Let  c  denote  the  edge  of  any  given  cube  ;  take  ^i  so  that 

OE:   EQ::c:c,,  or    OE'  :   W  -  «'  =  «i'- 
But  W  =  2 7a£^' ;  th eref ore,  Cj'  =  2c^ ; 

that  is,  ci  is  the  edge  of  a  cube  double  the  given  cube  in 
volume.  In  like  manner,  by  taking  CB  =  mr,  we  can  find 
the  edge  of  a  cube  m  times  the  given  cube  in  volume. 


HIGHER    PLANE    CURVES. 


211 


199.  The  Conchoid  of  Nicomedes.  The  Conchoid  is  the 
locus  of  a  point  P  such  that  its  distance  from  a  fixed  line 
XX',  measured  along  the  line  through  F  and  a  fixed  point 
A,  is  constant.  A  is  the  Pole,  .\'A''  the  Directrix,  and  the 
constant  distance  BP,  denoted  by  b,  is  the  Parameter. 


P>^ 

Y 

JM    

\k'  m 

0    /r 

~^_ 

Y 

A 

Y 

-■■B 

Fig.  80. 

To  construct  the  conchoid  by  points,  through  A  draw  any 
line  AP  cutting  XX'  in  P.  La}'  off  PP^^h  on  both  sides 
of  XX'.  In  like  manner  locate  the  points  P',  and  any 
number  of  others,  and  trace  the  branches  through  them. 

To  find  the  equation  of  the  curve  referred  to  XX'  and  a 
line  through  A  perpendicular  to  A'A'' ;  let 
0M=  x,  MP  =  y,  AO  =  a. 

Now,  AB  :  BP::  RM :  MP, 

and  BM=  ^'rF  -  HlF  =  ^iW^f. 

Therefore,   x  \  y-\-<t  ■.:  ■\/lr'  — if  :  //. 
Therefore,  x'^y^  =  {y  +  af- {b^  -  y'),  [49] 

which  is  the  equation  of  both  branches  of  the  conchoid.     A 
discussion  of  [49]  leads  to  the  following  conclusions  : 

(i)    The  curve  lies  between  the  lines  //  =  ^'  and  //^  —h. 

(ii)   The  curve  is  symmetrical  with  respect  to  the  axis  of  y. 


212 


ANALYTIC    GEOMETRY. 


(iii)  The  axis  of  x  is  an  asymptote  to  each  branch  of  the 
curve. 

If  6  >  a,  the  lower  branch  has  an  oval  or  loop,  as  in  the 
figure. 

If  i  =  a,  the  lower  branch  passes  through  A  and  is  some- 
what like  that  in  the  figure,  without  the  loop  below  A. 

If  6  <  a,  the  upper  and  lower  branches  are  like  the  dotted 
lines  in  the  figure. 

If  a  =  0,  the  conchoid  becomes  a  circle. 

CoR.  1.    If  A  is  the  pole  and  A  Fthe  polar  axis,  we  have 
p  =  AB'±B'F'  =  asec  6  ±h, 
which  is  the  polar  equation  of  the  conchoid. 

The  conchoid,  invented  by  Nicomedes,  a  Greek  mathema- 
tician of  the  second  century  b.c,  was,  like  the  cissoid,  first 
formed  for  solving  the  problem  oi  finding  ttvo  mean  propor- 
tionals or  duplicating  the  cube.  It  is  more  readily  applicable, 
however,  to  the  trisection  of  an  angle,  a  problem  not  less 
celebrated  among  the  ancients. 


Cor.  2.  To  trisect  any  angle,  as  CAP  (Fig.  81),  on  ^C 
lay  off  AB  any  length  ;  through  B  draw  BK  perpendicular 
to  AP,  and  take  KF  =  2AB.  Construct  a  conchoid  with 
^  as  a  pole,  XX'  as  a  directrix,  and  KP  as  a  parameter. 


HIGHER    PLANE    CURVES.  213 

At  R  erect  a  perpendicuhir  to  XX'  intersecting  the  cou- 
choid  in  D  ;  then  DA  will  trisect  the  angle  RAF. 

For  bisect  DB  in  S ;   then 

RS  =  SD  =  iKF  =  AR. 

Therefore,  Z  RAS=  Z  ^'S'^  =  2Z  RDS=  2Z  DAP. 
Therefore,  Z  DAP  =  iZ  RAF. 


/ 


200.  The  Lemniscate  of  Bernoulli.  The  Lemniscate  is 
the  locus  of  the  intersection  of  a  tangent  to  a  rectangular 
hyperbola  with  the  perpendicular  to  it  from  the  centre. 

To  find  its  equation  we  proceed  as  follows  : 

The  equation  of  the  tangent  to  the  equilateral  hyperbola 
x^  —  y^  =  ci^  at  the  point  (xi,  i/{)  is 

a'la;— yi?/  =  «^-  (1) 

The  equation  of  the  perpendicular  from  the  origin  to  (1)  is 

y^-^:r,  or  -  =  -^.  (2) 

Xl  Xi  i/i 

Solving  (1)  and  (2)  for  x^  and  v/i  by  multiplying  each 
term  of  (1)  by  one  of  the  members  of  (2),  we  obtain 

a"x  a'i/ 

X  +  V  ==  —  = 

^1         y\ 

^,         .                       ti^x  ahi 

Therefore,   cr,  =  ^r-, — 7/  ?/i  ^ ;— r — ^• 

But,  since  (a^i,  y^)  is  on  the  hyperbola,  we  have  Xx — y\=^o^\ 
hence,  by  substitution,  we  obtain 

(^2  +  ^•2)2  =  „2(ar;2  _  2,2)^  ^5QJ 

which  is  the  equation  sought. 


214 


ANALYTIC    (ilOoMlCTHV. 


From  \_iiO^  it  follows  tliat  the  curve  is  syiimietrical  with 
respect  to  both  axes.  The  form  of  the  curve  is  given  in 
Fig.  82. 


Fig.  82. 


CoR.  1.    Substituting  p  ens  6  for  x,  anrl  p  sin  6  for  y  in 
[50],  and  remembering  that  cos'^  ^— sin^  ^=cos  20,  we  obtain 


p^:=a^  cos  20 
as  the  polar  equation  of  the  lemniscate. 


(2) 


CoR.  2.    From  (2),  p  =  ±a  Vcos  20. 

Hence,  when  0=^0,  p  =  ±a.  AsO increases  from  0°  to 45°, 
p  changes  from  ±  «  to  ±  0,  and  the  portions  in  the  lirst  and 
third  quadrants  are  traced.  As  0  increases  from  45°  to  135°, 
cos  20  is  negative  and  p  is  imaginary.  As  0  increases  from 
135°  to  180°,  p  changes  from  ±  0  to  ±  a,  and  the  portions 
in  the  second  and  fourth  quadrants  are  traced.     Therefore, 

(i)  The  curve  consists  of  two  ovals  meeting  at  the  pole  0. 
(ii)  The  tangents  to  the  curve  at  0  are  the  asymptotes 
to  the  equilateral  hyperbola. 


HIGHER  PLANP:  CURVES. 


215 


201.    The  Witch  of  Agnesi.     Let    17/  be  a  tangent 
to  the  circle  OKYnt  the  vertex  oi'  the  diameter  OY ;  let 


OB  be  any  line  from  0  to  Ylf  cutting  the  circle  in  K; 
produce  the  ordinate  I>K,  and  make  DP'^^YE;  then  the 
locus  of  P  is  the  Witch. 

To  find  its  equation,  let  the  tangent  OX  and  the  diameter 
or  be  the  axes;  let  0F=2randPbe  any  point;  then 
OD=ij,  DP  =  x,  and 

OD  :  OY  ::  I)K  :  DP{=  YE),  (1) 

9,  '~ 


or  y  -.'Zr  ::  \ly(2r  —  y)'-x. 

Therefore,   oc-y  =  4r-(2/-  —  y) 
is  the  equation  of  the  Witch. 


[51] 


CoR.  1.    Since  a;  =  ±  2j'  V(2/ — y)  -r-y,  it  follows  that 
(i)  The  curve  is  symmetrical  with  respect  to  the  axis  of  y- 
(ii)  The  curve  lies  between  the  lines  y=0  and  y  =  2r. 
(iii)  The  axis  of  x  is  an  asymptote  to  eacli  infinite  branch. 

CoR.  2.  From  (1)  it  follows  that  corresponding  abscissas 
of  the  circle  and  the  Witch  are  proportional  to  the  ordinate 
and  the  diameter  of  the  circle. 

The  Witch  was  invented  by  Donna  Maria  Agnesi,  an 
Italian  mathematician  of  the  eiQ:hteenth  oontnrv. 


216 


ANALYTIC    GEOMETRY. 


202.  The  Cycloid.  A  Cycloid  is  generated  by  a  point  P  in 
the  circumference  of  a  circle  RFC,  which  rolls  along  a  right 
line  OX.  The  curve  consists  of  an  unlimited  number  of 
branches,  but  a  single  branch  is  usually  termed  a  cycloid. 
The  right  line  0X\?>  called  the  Base ;  the  rolling  circle  RFC 
the  Generatrix ;  and  P  the  Generating  Point.  If  0K=^  KX, 
the  perpendicular  KH  is  the  Axis,  and  JI  the  Highest  Point. 


OM 


To  find  the  equation  of  the  curve,  first  take  as  the  axis 
of  X  the  base  OX,  and  as  the  origin  the  point  0,  where 
the  curve  meets  the  base.  Let  r  denote  the  radius  of  the 
generatrix  RPC,  and  $  the  angle  PCR;  then  arc  PR 
equals  OR  over  which  it  has  rolled,  and  6  ^=  a,vG  PR -^r. 
Denote  the  coordinates  of  P  by  x  and  y  ;  then 

X  =  0M=  OR  —  MR  =  arc  PR  —  PN=  r6  —  r  sin  d. 
y  =  MP  =  RC  —  NC=r—  r  mse. 
Therefore, 

X  =^r  (6  —  sin  6) 
y=^r  (1  —  cos  6) 
Equations  (1)  taken  as  simultaneous  are  the  equations  of 
the  cycloid. 

To  eliminate  6  between  these  equations,  from  the  second, 
we  obtain 


(1) 


HIGHER    PLANE    CURVES.  21' 


cos  0  = :  therefore,  sin  ^  =  ± — — : 

/•  /• 

and  vers  ^=ri —cos  ^1  =  - ;  or^  =  vers'^-- 

L  -*      r  r 

Substituting  these  values  of  6  and  sin  9  in  the  first  of 

equations  (1),  we  have 

x  =  r  vers  '^  a?  y.  T  ^2ri/  -  yS  [52] 

the  equation  of  the  cycloid  in  the  more  common  form. 

In  the  value  of  sin  6,  and  therefore  in  equation  [52], 
the  upper  or  lower  sign  is  used  according  as  ^  <  or  >  tt  ; 
that  is,  according  as  the  point  is  on  the  first  or  second  half 
of  a  branch. 

From  1/  =  r  (1  —  cos  $)  it  follows  that  the  locus  lies 
between  the  lines  i/  =  0  and  ?/  =  2r. 

For  ?/  =  0  in  [52],  x  =  0,  ±  2iri',  ±  47rr, ;    hence,  the 

locus  consists  of  an  unlimited  number  of  branches  like 
OHX,  both  to  the  right  and  to  the  left  of  0  Y. 

203.  Let  the  highest  point  0  (Fig.  85)  be  taken  as  the 
origin,  and  OX  parallel  to  the  base  as  the  axis  of  x;  then 
OY,  the  axis  of  the  curve,  will  be  the  axis  of  y.  Let  6 
denote  the  angle  HCK. 

The  point  K  was  at  Y  when  P  was  at  0,  and  arc  KH= 
YH.     Hence, 

X  =  0M=  Yir+  BP  =  re  +  r  sin  6. 

y  =  -  MP  =  —  XC  +  BC=  —  r  -\-  r  cos  $. 

Hence,  the  equations  are 

a-  —  ;■  (^  +  sin  6)  ")  ^^^ 

y=^r  (cos  ^  —  1)  )  ' 


or  a;  =  r  vers  ^  (  — •-  \  ±  V—  2rij  —  y\  (2) 


218 


ANALYTIC    GEOMETRY, 


R  Y 

Fig.  85. 

Tlie  invention  of  the  cycloid  is  usually  ascribed  to 
Galileo.  After  the  conies,  no  curve  has  exercised  the 
ingenuity  of  mathematicians  more  than  the  cycloid,  and 
their  labors  have  been  rewarded  by  the  discovery  of  a 
multitude  of  interesting  properties.  Thus,  the  lengtli  of 
the  branch  ROA  is  eight  times  the  radius,  and  the  area 
BOA  is  three  times  the  area  of  the  s:eneratins[  circle. 


Exercise  41. 

1.  Prove  that  the  cissoid  is  the  locus  of  the  intersection 
of  a  tangent  to  the  parabola  if^=  —  8ra;  with  the  perpen- 
dicular to  it  from  the  origin. 

By  §  99,  the  equation  of  a  tangent  to  if^  —  Srx  is 

(1) 


y  =  mx  ■ 


— ) 
m 


(2) 


and  the  perpendicular  to  it  from  (0,  0)  is 

y  = X  :  therefore,  ?m  = 

m  y  ^ 

Eliminating  m  between  (1)  and  (2),  we  obtain 

a-' 

•'^"=27^- 

2.  At  the  centre  of  any  circle  C  (Fig.  86),  erect  CH  _L 
to  the  diameter  OA';  and  on  A'O  produced  layoff  0^4  = 
OC^i'.  Let  LQii  be  a  rectangular  ruler  of  which  the  leg 
QR  equals  2r.     If  the  ruler  is  moved  so  that  the  leg  LQ 


HIGHKIl    PLANK    CURVES. 


219 


passes  through  A,  while  the  end  li  sli.Ies  along  C'll,  ])rove 
that  the  locus  of  P,  the  middle  point  of  Qli,  is  a  cissoid. 


Fig.  8fi. 


Let  OA^be  the  axis  of  a^,   0  the  origin,  and  AQK  any 
position  of  the  ruler  ;  then  OM^x,  MP^=^y, 

EQ  =  MN^  CM=  X  —  r,     A X=  2 0M=  2x. 
EP^  =  PQ"  —  JiJif  =  r'  —  (.T  -  ry  =  2rx  —  x'\    (1 ) 


y  =  MP=NQ-\-EP. 
NQ-.AN.-.EQ.EP, 


or  NQ  :  2.r  : :  .r  —  r  :   \l2rx  —  x' 

From  (1),  (2),  and  (,']),  we  obtain 

\j2rx  —  .1 


(-0 
(3) 


y 


+  yj2rx-x% 


or  !f- 


:^r  —  X 

This   method  of  deseribing    tlie    oissoid   by  continuous 
motion  was  invented  by  Sir  Isaac  Newton. 


220  ANALYTIC    GEOMETRY. 

3.  In  Fig.  79  prove  that  jVS  and  OJV  are  two  mean  pro- 
portionals between  OM  and  iV7 ;  that  is,  prove 

OM-.NS::  ON:  NI. 
NS''  =  NXX  0N=  OMX  ON. 
The  right  line  01  will  pass  through  K;  hence, 

Z  OJN^  Z  VOI=  ^  arc  0K=  ^  arc  SX 
^ZSOX; 
therefore,         NS:  ON::  ON:  NI,  etc. 

4.  If  in  the  lemniscate  (Fig.  82)  0F'=0F=^a^2, 
prove  that  FF  X  F'F  is  constant,  F  being  any  point  on  the 
curve;  and  hence  that  the  lemniscate  may  be  defined  as 
the  locus  of  a  point  the  product  of  whose  distances  from 
two  fixed  points  is  constant. 

5.  Construct  the  logarithmic  curve  y^=a^,  or  x^log^y. 
Prove  that  every  logarithmic  curve  passes  through  the 
point  (0,  1),  and  has  the  axis  of  x  as  an  asymptote. 

6.  The  Trochoid  is  the  curve  traced  by  any  point  in  the 
radius  of  a  circle  rolling  on  a  right  line.  If  r  denotes  the 
radius  of  the  circle,  h  the  distance  of  the  generating  point 
from  its  centre,  and  Q  denotes  the  same  angle  as  in  §  202, 
show  that  the  equations  of  the  trochoid  are 

x  =  r6  —  b  sin  $  >  ,.y. 

y=^r    —  b  cos  6  ) 
When  b<ir,  the  trochoid  is  called  the  Prolate  Cycloid; 
and  when  b>  r,  the  Curtate  Cycloid.      When  b  =  r,  the 
curve  is  the  cycloid. 

Spirals. 

204.  A  Spiral  is  the  locus  of  a  point  whose  radius  vector 
continually  increases,  or  continually  decreases,  while  its 
vectorial  angle  increases  (or  decreases)  without  limit. 


HIGHER    PLANK    CURVES. 


221 


A  Spire  is  the  portion  of  the  spiral  traced  during  one 
revolution  of  the  radius  vector. 

The  Measuring  Circle  is  the  circle  whose  centre  is  the 
pole  and  whose  radius  is  the  value  of  p  when  6^2Tr. 

205.  The  Spiral  of  Archimedes.  If  the  radius  vector 
of  a  moving  point  has  a  constant  ratio  to  its  vectorial 
angle,  that  is,  if 

p  =  ce,  ■      (1) 

the  locus  is  the  Spiral  of  Archimedes. 
From  equation  (1),  we  have 

when^  =  0,  Itt,    |7r,    |7r,    ir,     |7r,    |7r,     |7r,     27r,    |7r,    

p=0,  \itC,  lire,  fTTC,  ttC,  ^ttC,  &TrC,  JttC,  2TrC,  ^wC,  


Hence,  to  construct  the  Spiral  of  Archimedes,  draw  the 

radial  lines  OH',  OA,  OB,  OG,  including  angles  of  ^n; 

on  these  lay  off  Oa  =  J  7rr,  Ob  =  |7rr,  Oc  =  ^irc, ,  OH^  27rc, 

and  trace  the  first  spire  OabcdefgH  through  these  points. 
Any  number  of  other  spires  are  easily  constructed  by  noting 
that  the  distance  between  two  spires,  measured  on  a  radius 
vector,  is  equal  to  2irc.  Thus,  by  taking  a  A.  bB,  cC,  dD,  eE, 
fF,  gG,  HH',  each  equal  to  2irc,  we  obtain  points  of  the 
second  spire  ;  and  so  on.  Any  number  of  a(^ditional  radial 
lines  may  be  drawn  to  locate  ]ioints  in  tlie  curve. 


222 


ANALYTIC    GEOMETRY. 


The  spires,  being  everywhere  equally  distant  along 
radial  lines,  aie  said  to  be  parallel.  The  measuring  circle 
is  JIMS,  whose  radius  is  OH  or  2Trc. 

206.  The  Reciprocal  or  Hyperbolic  Spiral.  If  the  radius 
vector  of  a  point  varies  inversely  as  its  vectorial  angle, 
that  is,  if  pd=^c,  (1) 

the  locus  is  the  Reciprocal  Spiral. 

Since  p=^c^6,  we  have 
when    6  =  0,        lir,       lir,      ^tt,      tt. 


4  "  '        2  "  '         4"} 


(  V  V  V  r,  V  V        V 

p  =  CC,i^^       >    4  ^,    «  — ,    J   — ,    A  — ,    4  8  

ZtT  wTT  wTT  -iTT  ^TT  iiTT  ZtT    ZtT 

Hence,  the  radius  of  the  measuring  circle  is  c-f-27r,  and 
its  circumference  is  c. 


Fig.  88. 

To  construct  the  curve,  draw  the  radial  lines  OX,  Oa,  Ob, 
Oc,  Od,  Oe,  Of,  ();/,  including  angles  of  Jtt.  Take  OH^  e 
-^ Stt,  and  lay  off  Oa  =  8  X  OJf,  Ob ^ 4  X  OH,  Or  =f^X OH, 
0(1  =  2  X  OH,  etc.;  and  through  the  points  a,  b,  c,  d,  etc., 
trace  the  curve.  In  like  manner,  any  number  of  spires  may 
be  drawn.  From  (1)  it  is  evident  that  p  approaches  zero, 
as  0  approaches  infinity ;  that  is,  the  curve  continually 
approaches  the  pole  without  ever  reaching  it. 

Since  pd  =  e,  it  follows  that  the  arc  FK  described  with 
the  radius  vector  of  any  point  /'is  constant  and  equal  to  c. 


HIGH  KM    PLANK    CUUVKS. 


223 


Now,  as  p  appi'oaclies  infinity,  tiiis  arc  a,|)[n-();i(',lies  a  perpen- 
dicular to  OX.  Hence,  the  line  parallel  to  OX  at  the  distance 
of  c  above  it  is  an  as}- niptote  to  the  infinite  brancli  of  the 
spiral. 

207.    The  Lituus.     If  the  square  of  the  radius  vector  of  a 
point  varies  inversely  as  its  vectorial  angle,  that  is,  if 

irO  =  c, 
the  locus  is  the  Lituus. 


Fig.  80. 

Let  the  student  construct  tlie  curve  from  its  equation 
and  show  that 

(i)    The  curve  continually  approaelies  the  pule  without 
ever  reaching  it,  as  6  increases  withoiit  limit. 

(ii)    The  polar  axis  is  an  asymptote  to  the  infinite  branch. 

208.  The  Logarithmic  Spiral.  If  tlie  radius  vector  of 
a  point  increases  in  a  geometrical  ratio,  while  its  vectorial 
angle  increases  in  an  arithmetical  ratio ;  that  is,  if 

p  =  a^,  or  6=:log„p,  (1) 

the  locus  is  the  Logarithmic  SpiraL 

Since  p  =  l  when  6  =  0,  every  logarithmic  spiral  passes 
through  the  point  (1,  0). 

To  construct  a  logarithmic  s])iral,  let  r/  =  2;  then  p  =  2". 

When^  =  0,     1(=57.,r),     2(=  114.0°) '2^. 

P  =  l,     2  ,4  _       ,     ,     77.88. 


224  ANALYTIC    GEOMETRY. 

In  Fig.  90  let  XOb  =  57.3°,  XOc  =  114.6°,  Oa  =  l,0b  =  2, 
Oc  =  4:,  then  a,  h,  c  are  three  points  on  the  spiral.  As  6 
increases,  p  increases  rapidly,  but  it  becomes  infinity  only 
when  B  does ;  and  hence  only  after  an  infinite  number  of 


Fig.  90. 

revolutions.  As  6  decreases  from  zero,  p  decreases  from  unity. 
Since  p  approaches  zero  as  9  approaches  negative  infinity, 
the  curve  approaches  the  pole  without  ever  reaching  it. 

209.  The  Parabolic  Spiral.  If  in  the  equation  if  =  ^)x, 
the  values  of  x  are  laid  off  from  A  (Fig.  91)  on  the  circle 
AH,  and  those  of  y  on  its  corresponding  radii  produced,  the 
locus  of  the  point  thus  determined  is  the  Parabolic  Spiral. 

To  find  its  equation,  denote  the  radius  OA  by  r,  and  let 
P  be  any  point ;  then 

X  =  AmH  ^=  rO. 

y  =  HF=OF—OH=p  —  r. 


HIGHER    PLANE    CURVES.  225 

Siibstitutiug  these  values  of  x  and  //  in  y^  =  4^jx,  we  ob- 
tain the  polar  equation 


Fig.  91. 

The  curve  consists  of  two  branches  beginning  at  A ;  the 
one  determined  by  the  positive  values  of  y  is  an  infinite 
spiral  lying  entirely  without  the  circle;  the  other  branch 
passes  through  the  pole,  forms  a  loop,  and  passes  without 
the  circle  when  p  =  —  /•,  and  ^  =  ?•  -|-^j. 

Note.  Among  the  ancients  no  problems  were  more  celebrated  than 
the  "Duplication  of  the  Cube"  and  the  "Trisection  of  an  Angle." 
Hii^pocrates  of  Chios  reduced  these  two  problems  to  the  more  general 
problem  of  finding  two  mean  proportionals  between  two  given  lines. 
Thus,  if  c  is  the  edge  of  the  given  cube,  and  x  and  y  are  the  two  mean 
proportionals  between  c  and  2c,  we  have 
c :  X  =  X  :  y  =  y  :  2c. 

Therefore,  (~^  -  -  X  -  X  ^^- =  i,  or  x^  -  2c^ 

\x/       x      y      2c 

Hence,  x,  the  first  of  the  two  mean  proportionals  between  c  and  2c, 
is  the  edge  of  a  cube  double  the  given  cube  in  volume. 

After  years  of  vain  endeavor  to  solve  these  problems  by  the  right 
line  and  the  circle,  the  ancient  geometers  began  to  invent  and  study 
other  curves,  as  the  conies  and  some  of  the  higher  plane  curves.  The 
invention  of  the  conies  is  credited  to  Plato,  in  whose  school  their 
properties  were  an  object  of  special  study. 


PART   11. —  SOLID    GEOMETRY. 


CHAPTER   I. 


THE   POINT. 


210.  The  position  of  a  point  in  space  may  be  determined 
by  referring  it  to  three  fixed  })lanes  meeting  in  a  point. 
The  fixed  planes  are  called  Coordinate  Planes,  their  lines 
of  intersection  the  Coordinate  Axes,  and  their  common 
point  tlie  Origin.  In  what  follows  we  shall  employ  coordi- 
nate planes  that  intersect  each  other  at  right  angles. 

Let  XOY,  YOZ,  ZOX,  be  three  planes  of  indefinite  ex- 
tent intersecting?  each  other  at  risrht  anerles  in  the  lines 


A'A',  YY',  ZZ'.  These  coordinate  planes  are  called  the 
planes  xy,  yz,  zx,  respectively;  the  axes  XX\  YY\  ZZ^  are 
called  the  axes  of  x,  y,  z,  respectively;  and  their  common 
point  0  is  the  origin. 


TIIK    POINT.  227 

The  coordinate  i)laiies  divide  all  space  into  eight  portions, 
called  Octants,  wliich  are  numbered  as  follows:  The  First 
Octant  is  0-XYZ,  thi-  Secomi,  0-YX'Z,  the  Third,  0-X'Y'Z, 
the  Fourth,  0-Y'A'Z,i\ni  Fifth,  0-XYZ',  the  Sl.rth,  0-  YX'Z', 
the  Seventh,  0-X'Y'Z',  the  Fiffhth,  0-Y'XZ'.  The  lifth 
octant  is  below  the  first. 

Let  F  be  any  point  in  space,  and  througli  it  pass  three 
planes  parallel  respectively  to  the  three  coordinate  planes, 
thus  forming  the  rectangular  parallelopiped  F-OJtXM. 
The  position  of  F  will  be  determined  when  we  know  tiie 
lengths  and  directions  of  the  lines  LP,  QP,  NP.  These 
three  lines  are  called  the  Rectangular  Coordinates  x,  y,  z, 
of  the  point  P,  which  is  written  {i-,  ij,  z). 

A  coordinate  is  jjositive  when  it  has  the  direction  of  OX, 
OY,  or  OZ;  hence,  it  is  ner/atlve  when  it  has  the  direction 
of  OX',  OY',  or  OZ'. 

Thus  the  coordinate  x  is  positive  or  negative  according 
as  it  extends  to  the  right  or  to  the  left  from  the  plane  //.~; 
1/  is  positive  or  negative  according  as  it  extends  to  the  front 
or  to  the  rear  of  the  plane  zx;  and  z,  according  as  it  ex- 
tends upward  or  downward  from  the  ]ilane  x//.  Hence,  the 
octant  in  which  a  point  is  situated  is  determined  by  the 
signs  of  its  coordinates.  Since  the  first  octant  has  the  posi- 
tive directions  of  the  axes  for  its  edges,  tlie  coordinates  of 
a  point  in  the  first  octant  are  all  i)Ositive.  If  (a,  b,  e)  is  a 
point  in  the  first  octant,  the  corresponding  point 

in  the  second  octant  is  ( — a,  h,  r), 

in  the  third  octant  is  ( —  (t,  —  b,  c), 

in  the  fourth  octant  is  (k,  —  b,  r), 

in  the  fifth  octant  is  (a,  b,  —  r), 

in  the  sixth  octant  is  ( — a,  b,  — <•), 

in  the  seventh  octant  is  ( — a,  — b,  — c), 

in  the  eighth  octant  is  {(i,  — b,  — c). 


228  ANALYTIC    GEOMETRY. 

The  point  (x,  y,  0)  is  in  the  plane  xy. 
The  point  (x,  0,  0)  is  in  the  axis  of  x. 
The  point  (0,  0,  0)  is  the  origin. 

The  lines  OM,  OR,  OS,  or  OM,  MN,  NP,  may  be  taken 
as  the  coordinates  of  P,  for  they  have  the  same  length  and 
direction  as  LP,  QP,  NP,  respectively.  To  construct  P, 
(x,  y,  z),  we  take  OM^x,  draw  MN  parallel  to  OY,  take 
MN=y,  draw  NP  parallel  to  OZ,  and  take  NP  =  z. 

211.  The  Radius  Vector  of  a  point  is  the  line  drawn  to 
it  from  the  origin.  Thus,  OP  (=p)  is  the  radius  vector  of  P. 
From  the  rectangular  parallelopiped  in  Fig.  92,  we  have 


op"  =  Oil'  -f  MN"^  +  NP". 
Hence,  denoting  the  coordinates  of  P  by  x,  y,  z,  we  have 

That  is,  the  square  of  the  radius  vector  of  a  point  is  equal 
to  the  sum  of  the  squares  of  its  rectangular  coordinates. 

212.  By  the  angle  between  two  non-intersecting  straight 
lines  is  meant  the  angle  between  any  two  intersecting 
straight  lines  that  are  parallel  to  them.  Thus,  any  line 
parallel  to  OP  (Fig.  92)  makes  the  angles  XOP,  YOP,  ZOP 
with  the  axes  of  x,  y,  z,  respectively. 

The  angles  which  a  line  makes  with  the  positive  direc- 
tions of  the  coordinate  axes  are  called  its  Direction  Angles ; 
and  the  cosines  of  these  angles  are  called  the  Direction 
Cosines  of  the  line. 

The  direction  angles  of  a  line  are  always  positive  and 
cannot  exceed  ir,  or  180°. 

213.  Let  a,  fS,  y  denote  the  direction  angles  of  OP  (Fig. 
92),  or  any  line  parallel  to  it,  and  x,  y,  z  the  coordinates  of 
P,  and  let  p  =  OP;  then  evidently 


THE    POINT.  229 

ac=  p cos  a,  y  =  p  COS  p,  s  =  p  cos  y.  [54] 

Squaring  and  adding  [54],  and  substituting  in  [53],  we 
obtain 

cos2  a  +  cos2  p  +  cos2  7=1.  [55] 

That  is,  the  sum  of  the  squares  of  the  direction  cosines  of 
a  line  is  equal  to  unity. 

Cor.  Whatever  are  the  values  of  x,  y,  z  in  [54],  if 
each  is  divided  by  p,  or  Vx^  +  2/^  +  «^  the  quotients  are  the 
direction  cosines  of  the  radius  vector  of  the  point  {x,  y,  z). 

Hence,  if  any  three  real  quantities  are  each  divided  by  the 
square  root  of  the  sum  of  their  squares,  the  qiwtients  will  be 
the  direction  cosines  of  some  line. 

Exercise  42. 

1.  In  what  octants  may  (oc,  y,  z)  be,  when  x  is  positive  ? 
when ic  is  negative ?  when  y  is  positive?  when ?/ is  negative ? 
when  z  is  positive  ?  when  z  is  negative  ? 

2.  Inwhatoctantis(— 2,4,  6)?  (2,4,-3)?  (—2,4,-1)? 
(-2,-3,-1)?  (-2,-3,3)?  (2,-3,1)?  (2,-1,-3)? 
Construct  each  point. 

3.  In  what  line  is  {a,  0,  0)  ?  (0,  0,  c)  ?  (0,  b,  0)  ? 

4.  In  what  plane  is  {a,  b,  0)  ?   {a,  0,  c)  ?  (0,  b,  c)  ? 

5.  Find  the  length  of  the  radius  vector  of  (3,  4,  5), 
(2,  — 3,  —1),  (7,  — 3,  —5).  Find  the  direction  cosines  of 
the  radius  vector  of  each  point. 

6.  The  direction  cosines  of  a  line  are  proportional  to 
1, 2,  3  ;  find  their  values.    What  is  the  direction  of  the  line  ? 

7.  What  is  the  direction  of  the  line  whose  direction 
cosines  are  proportional  to  A,  B,  C?  Wliat  are  the  values 
of  its  direction  cosines  ? 


230  ANALYTIC    GEOMETRY. 

8.  Two  direction  aiii;los  of  a  lino  aro  C>0°  and  45°,  what 
is  tlie  third  ?  If  two  are  (U)°  and  30°,  wliat  is  tlie  third? 
If  two  are  135°  and  G0°,  what  is  the  third  ? 

214.  Projections.  The  2)roJectio7i  of  a  point  upon  a  right 
line  is  the  foot  of  the  perpendicular  from  the  point  to  the 
line  ;  or  it  is  the  intersection  of  the  line  with  the  plane 
throngh  the  point  perpendicular  to  the  line.  Thus  31,  JR.,  S 
(Fig.  92)  are  the  projections  of  the  point  P  upon  the  axes 
of  X,  y,  z,  respectively.  Here  and  in  the  following  pages, 
by  projection  is  meant  the  orthogonal  projection. 

The  2^^'ojection  of  a  limited  right  line  on  another  right 
line  is  the  part  intercepted  between  the  projections  of  its 
extremities.  Thus,  031,  OB,  OS  (Fig.  92)  are  the  projec- 
tions of  OF  on  the  axes  of  x,  y,  z,  respectively. 

That  is,  the  coordinates  of  anypnhU  are  the  i^rojeetlons  of 
its  radius  vector  o7i  the  three  axes. 

215.  The  projections  of  any  line  PQ  on  parallel  lines  are 
equal  ;  for  these  projections  are  parallel  lines  included 
between  parallel  planes  through  P  and  Q.  Now  tlie  projec- 
tion of  any  straight  line  on  another  that  passes  tlirough  one 
of  its  extremities  is  evidently  equal  to  the  })roduct  of  its 
length  into  the  cosine  of  their  included  angle. 

Hence,  the  projection  of  any  limited  strait/ht  live  on  any 
other  straiyht  line  is  equal  to  its  length  nmdtiplied  hy  the 
cosine  of  the  angle  between  the  lines. 

216.  Let  AT)  (Figs.  93,  94)  be  a  straight  line,  and  ABCB 
any  broken  line  in,  space,  connecting  the  points  A  and  D,  and 
let  A\  B',  C",  D'  be  the  projections  of  A,  B,  C,  D  upon  OX, 
whose  positive  direction  is  OX.  Denote  the  angle  RADhy 
<!>,  the  lengths  of  the  lines  ^47?,  BC,  CD  by  l^,  l^,  4,  and  the 
angles  which  they  make  with  the  positive  direction  of  OX 
by  ai,  tta,  Us ;  then  in  Fig.  93   we  have 


TIIK     I'OINT. 


231 


A'jy  =  .17/'  +  7;'c"  +  c'lf. 

Therefore,   AI>vms^=Ii  cosai  +  /a  cos  o.^  +  ^3  cosa^.  [5(j] 
111  Fig.  04,  CD'  is  negative  ;  but  ^3  cos  a,  is  also  negative, 

since  ug,  or  SCV,  is  obtuse  5  hence  foriuuhi  [oG]  holds  true 

in  all  cases. 

D  ]) 


B'         C     D'    '"     "     A'  D'      D' 

Fig.  93.  Fig.  94. 

That  is,  the  ulgebralc  sum  of  the  ijrnjertiDnx  on  a  r/iren  line 
of  the  jifU'ts  of  (1)11/  broken  line  connecthuj  aiii/  two  points  is 
equal  to  the  projection  on  the  same  line  of  the  straiijht  line 
joining  the  same  two  points. 

2,\1 .  To  find  the  angle  between  two  straigJit  lines  in  terms 
of  their  direction  cosines. 


FiR.  95. 


Let  OP  and  OF' be  ])aralh'l  respectively  to  any  two  given 
lines  in  space.     Let  6  denote  their  included  angle,  and  u.  /?.  y 


232 


ANALYTIC    GEOMETRY. 


and  a,  (3',  y'  their  direction  angles,  respectively.  Let  OM, 
MN,  NP'  be  the  coordinates  of  P' ;  then  the  projection  of 
OP'  on  OP  equals  the  sum  of  the  projections  of  OM,  MN, 
NP'  on  OP  ;  that  is, 

OP' cos  6  =  OM  cos  a  +  MN  COS  (i  +  NP'  cos  y. 
But        0M=  OP' cos  a',  MN=OP'  cos  ^',  NP'=^  OP' cos y'. 
Hence,  OP'  cos  6  =  OP'  cos  a'  cos  a  +  OP'  cos  /3'  cos  (3 

-\-  OP'  cos  y'  COS  y, 

or       cos  9  =  cos  a  cos  a' +  cos  p  cos  P' +  cos  7  cos  y,       [57] 

which  is  the  required  formula. 

'  218.  To  find  the  distance  between  two  points  in  terms  of 
their  coordinates. 


Fig.  96. 

Let  Pi  be  the  point  (cci,  y^,  z^),  and  Pg  the  point  {x^^y^,  «2). 
Through  P^  and  Pj  pass  planes  parallel  to  the  coordinate 
planes,  thus  forming  the  rectangular  parallelepiped  whose 
diagonal  is  P1P2,  and  whose  edges  PiL,  LK^  KP^  are 
parallel  to  the  axes  of  x,  y,  z,  respectively. 

Then  1\P'  =  1\V  ^Tk'  +  KPI  (1) 


THE    POINT. 


233 


Now  P^L  is  the  difference  of  the  distances  of  P^  and  Pg 
from  the  plane  yz,  so  that  P^L=^x^  —  x-^.  For  like  reason, 
LK=  1/2  —  yi,  and  KP^  =  ^2  —  ^i-  Hence,  denoting  the  dis- 
tance P1P2  by  D,  we  have,  by  substituting  in  (1), 

D  =  V(X2  -  iCiY'  +  {»/2  -  yif  +  (^2  -  si)S  [58] 

which  is  the  required  formula. 

Cor.  1.  Since  PiL,  LK,  KP^  are  equal  to  the  projections 
of  the  line  P1P2  on  the  coordinate  axes,  it  follows,  from 
[58],  that 

The  square  of  any  line  is  equal  to  the  sum  of  the  squares 
of  its  2)roJections  011  the  axes. 

CoR.  2.  If  a,  ^,  y  are  the  direction  angles  of  the  line 
P1P2,  we  have 

X2  —  Xi^  D cos  a,  ?/2  —  yi=:D cos  /?,  «2  —  «i  =  -E> cos  y. 

219.  Polar  Coordinates.  Let  XOY  be  a  fixed  plane, 
OX  a  fixed  line  in  it,  and  OZ  a  perpendicular  to  it  at  the 
fixed  point  0.     To  P,  any  })oint  in  space,  draw  OP,  and 


through  OP  pass  a  plane  perpendicular  to  A'OY,  intersect- 
ing the  latter  in  OJV;  then  tlie  distance  OP  and  the  angles 
ZOP  and  WON  determine  the  point  P,  and  are  called  its 


234  ANALYTIC    GEOMETRY. 

Polar  Coordinates ;  Ol',  denoted  b^^  p,  is  the  Radius  Vector ; 
and  tlie  angles  ZOP  and  MON,  denoted  by  6  and  <^,  re- 
spectively, are  the  Vectorial  Angles.  The  point  F  is 
written  (p,  6,  cf>).  <^  determines  the  plane  ZOX,  6  determines 
the  line  OF  in  that  plane,  and  p  locates  P  in  OP. 

CoR.  If"  XOY  is  a  riglit  angle,  the  rectangular  coordi- 
nates of  P  are  OM,  MX,  XP.  To  express  these  in  terms 
of  the  polar  coordinates  of  P,  we  have 

X  =  OM  =  O^'^cos  <^  =  OP  sin  6  cos  4>^P  ^i^^  ^  ^''^^  ^■ 
y  =  MX=  OX&\\ifi>=^  OP  sin  ^  sin  <^  = /J  sin  ^  sin  ^. 
z  =  XP^  OP  cose=p  cos  9. 
We  readily  obtain  also 


tan  Q  = !-^  ^    tan  <i>=~- 

z  X 

220.  The  Projection  of  a  p(jint  on  a  plane  is  the  foot  of 
the  perpendicular  from  the  point  to  the  plane.  The  per- 
pendicular itself  is  the  Projector  of  the  point.  Thus,  the 
point  iV(Fig.  97)  is  the  i)rojection  of  P  on  the  plane  xy, 
and  PTVis  its  projector. 

The  jjt'ojection  of  a  limited  straight  line  on  a  jilane  is  the 
straight  line  joining  the  projections  of  its  extremities.  The 
Inclination  of  a  line  to  a  plane  is  the  angle  it  makes  with 
its  projection  on  that  plane.  The  projection  of  a  limited 
line  is  evidently  equal  to  its  length  multiplied  by  the 
cosine  of  its  inclination.     Thus,  0X=  OP  cos  XOP. 

The  p7-ojectio7i  of  any  curve  upon  a  plane  is  tlie  locus  of 
the  projections  of  all  its  points.  The  Projecting  Cylinder 
of  a  curve  is  tlie  locus  of  the  projectors  of  all  its  ])oints. 
In  the  case  of  a  right  line  this  locus  is  the  Projecting  Plane. 


THK    POINT.  2.35 

Ex^cise  43. 

1.  Find  tlie  distance  between  the  point.s  (1 , 2, 3), (2, 3, 4); 
(2, 3, 4),  (3,  4,  5);  (1,  2,  3),  (3,  4,  5). 

2.  Prove  that  tlie  triangle  formed  by  joining  tlie  three 
points  (1, 2,  3),  (2,  3, 1),  (3, 1, 2)  is  equilateral. 

3.  The  lengths  of  the  projections  of  a  line  on  the  three 
coordinate  axes  are  3,  4,  />,  respectively;  fin<l  the  length  of 
the  line. 

4.  Find  the  direction  cosines  of  the  radius  vector  of 
the  point  ( — 3,  — 4,  5). 

5.  Wliat  lines  have  direction  cosines  proportional  to  3, 

—  2,  —  5  ?     Find  the  values  of  these  direction  cosines. 

6.  Find  the  angle  between  two  straight  lines  whose  di- 
rection cosines  are  proportional  to  1,  2,  3  and  2,  3,  6,  respec- 
tively. 

7.  Find  the  angle  between  two  straight  lines  whose  di- 
rection cosines  are  proportional  to  1,  2,  3  and  5,  — 4,  1, 
respectively. 

8.  Find  the  polar  coordinates  of  the  point  ( V3, 1,  2  V3). 

9.  Find  the  rectangular  coordinates  of  (4,  ^tt,  -^tt). 

10.  \i  (.r,?/,.~)bisects  tlie  line  joining  (^'i,yi,r:i)to(.r^,  t/.i.  :^o), 
prove  that        x  =  ^(xy  +  .r^),  y  =  hiiu  +  Z/a),  '-  =  i(-i  +  ■-2)- 

11.  If   (x,  1/,  z)   divides    the    line  joining  (xy,  //,,  Zi)  to 
(x^,  1/2:  ^2)1  ii^  the  ratio  viy  :  vi„,  prove  that 

irioXy  -}-  niyXo  ?/?oVi  +  viyf/n  nu~i  -\-  miS^ 

X  =  — = — . -^     y  =  —=^ 1 ^5     ,-  =  — — , 

12.  Find  the  coordinates  of  the  point  that  divides  the 
line  joining  (3,  —2,  4)  and  (1,  3,  —  2)  in  the  ratio  1  :  3. 

13.  Find  the   point  that  divides  the  line  joining  ( — 2, 

—  3,-1)  and  (—5,  —2,  4)  in  the  ratio  0  :  2. 


CHAPTER   II. 

THE  PLANE. 

221.    To  find  the  equation  of  a  plane  in  terms  of  the  length 
of  the  perpendicular  from  the  origin,  and  its  direction  cosines. 


Fig.  98. 

Let  OF  be  the  perpendicular  to  the  phme  ABC  from  the 
origin  0;  denote  its  length  by^,  and  its  direction  angles  by 
a,  )8,  y.  Let  P  be  any  point  in  the  plane,  OP  its  radius  vec- 
tor, and  OM,  MN,  NP  its  coordinates,  x,  y,  z.  Then  the 
projection  of  OP  on  OF  is  equal  to  the  sum  of  the  projec- 
tions of  OM,  MN,  NP  on  OF.  But  as  the  plane  is  perpen- 
dicular to  OF,  p  is  the  projection  of  OP  on  OF;  and  the 
projections  of  OM,  MN,  NP  on  OF  are  respectively  x  cos  a, 
y  cos  (B,  z  cos  y  ;  hence, 

j7cosa  +  2/cosP  +  scos-y  =  2>>  [59] 


THE    PLANE.  237 

which  is  the  equation  required.     Equation  [59]  is  called 
the  normal  equation  of  a  plane. 

Cor.  1.    When  the  plane  is  perpendicular  to  one  of  the 

coordinate  planes,  the  plane  xyiov  example,  Oi^lies  in  the 

plane  xy ;  hence,  y  =  •j-ir,  cos  y =0,  and  equation  [59]  becomes 

X  co^  a-\-  y  CO?,  [i^p.  (1) 

CoR.  2.  When  the  plane  is  parallel  to  one  of  the  coordi- 
nate planes,  as  the  plane  yz,  Oi^lies  in  the  axis  of  x  ;  hence, 
cos  a  =  l,  cos  ^  =  0,  cos  y==0,  and  [59]  becomes 

x=p.  (2) 

CoR.  3.  Since  OF  is  perpendicular  to  the  plane  ABC, 
and  OX  to  YOZ,  the  dihedral  angle  J-i?C-0  =  angle  FOX. 
For  like  reason,  B-CA-0  =  FOY,  and  C-BA-0  =  FOZ. 

222.  The  lociis  of  every  equation  of  the  first  degree  between 
three  variables  is  a  plane. 

A  general  form  embracing  every  equation  of  the  first 
degree  between  x,  y,  z  is 

Ax  +  By+Cz  =  D,  (1) 

in  which  D  is  positive. 

Dividing  both  members  of  (1)  by  V--1"  +  B^  +  C'\ 
we  obtain 

A  ,  B  ,  C 

'J- 


=    ,        ^  ,  (2) 

in  which  the  coefficients  of  x,  y,  z  are  the  direction  cosines 
of  some  line  (§  213,  Cor.).  Thus  (2)  is  in  the  form  of  [59] 
§  221  ;   hence,  the  locus  of  (2),  or  (1),  is  a  plane. 

CoR.   1.    The  length  of  the  perpendicular  from  the  origin 
upon  plane  (1)  equals  the  second  member  of  equation  (2),  and 


238  ANALYTIC    GEOMETRY. 

the  direction  cosines  of  tliis  pfrpcndicular  are  the  coeflficients 
in  (2)  of  X,  I/,  z,  resi^ectively.  These  direction  cosines  are 
evidently  proportional  to  A,  B,  C. 

Hence,  to  construct  equation  (1),  draw  tlie  radius  vector  of 
the  point  {A,  B,  C)  ;  the  plane  perpendicular  to  this  line  at 

the  distance     , from  theoritrin  is  the  locus  of  (1 ). 

Cou.  2.  To  reduce  any  simple  equation  to  tlie  ttur/nal 
form,  put  it  in  the  form  of  A.r  +  Jli/-\-  Cz  =  I),  in  which  U  is 
positive,  and  divide  both  niembers  by  '\/A^-\-  J J'^-\- €''■'. 

CoR.  3.  If  a  simple  equation  contains  only  two  variables, 
its  locus  is  perpendicular  to  the  corresponding^  coordinate 
plane  ;  if  only  one  variable,  its  locus  is  perpendicular  to 
the  corresponding  coordinate  axis. 

223.  To  find  tlie  e(/'Ucftlo)i  of  a  plane  in  terms  of  its  inter- 
cepts on  the  axes. 

Let  a,  b,  c  denote  respectively  the  intercepts  on  the  axes 
of  the  plane  whose  equation  is 

Ax-\-n!/+Cz  =  B.  (1) 

Making  ?/  =  ,^•=:0,  and  therefore  x^^a,  (1)  becomes 

Aa  =  IJ,  or  A=:  D-T-  a. 
Making  x  =  z^^  0,  and  therefore  //  =  b,  (1)  becomes 

m=D,  or  B^D-^h. 
Making  a;  =  //:=  0,  and  therefore  ,t'  =  (",  (1)  becomes 

Cc  —  I>,  or  C—D^c. 
Substituting  these  values  in  (1)  and  dividing  by  Z>,  we  have 

^+^^+5=1,  [60] 

which  is  the  required  equation.  Equation  [60]  is  called 
the  sijniinetrical  equation  of  a  plane. 


THK     PLANK.  239 

224.    To  find  the  (nKjJe  hetween  any  two  planes. 
The  angle  included  lietween  the  two  planes 
Ax-\rB'y^-Cz  =  D\ 
Ax-\-By+  Cz  =  D, 
is  evidently  equal  to  the  angle  included  between  the  perpen- 
diculars to  them  from  the  origin.     l>ut  the  direction  cosines 
of  these  perpendiculars  are  respectively  (§  222,  Cor.  1), 
ABC 


ylA'  +  B-  +  V'    V.i-  +  B""  -[-  C   ^/A'  +  B'  +  C' 
A'  B'  C" 


Substituting  these  values  in  [;">"],  we  have 

AA'^BIi'^CC 

cos  0  = 


[Gl] 


in  which  6  erpials  tlie  angle  included  between  the  planes. 

Cor.  1.  If  the  planes  are  iiaralld  to  each  other,  6^0 
and  cos  ^  =  1.  Putting  cos  6=^1  in  [<)1],  clearing  of  frac- 
tions, squaring,  transposing,  and  uniting,  we  obtain 

{AB'  -  BA 'f  +  {A  a  —  CA')'  -\-(BC-  CB'f  =  0. 

Each  term  being  a  square,  and  therefore  ])ositivt\  this 
equation  can  be  satisfied  only  when  each  term  equals  zero, 
giving  us 

AB'=BA,  AC"=  CA',  BC'=  CB\ 
^_B^_C 
^^         A~  B'~  C' 

Hence,  if  two  planes  are  parallel,  the  coefficients  of  x,  ij,  z, 
in  their  equations,  are  proportional,  and  eonversely. 

Cor.  2.  If  the  planes  are  2)erpendiciilar  to  each  other, 
cos  ^  =  0  ;  and  hence, 

AA'-\-BB'+CC'  =  0, 
and  converseli/. 


240  ANALYTIC    GEOMKTKY. 

225.  To  find  the  perpendlcidar  distance  of  a  given  'point 
from  a  given  pUnie. 

Let  the  equation  of  the  given  plane  be 

X  cos  a.-\-y cos /8  +  «  cos  y  =-p,  (1) 

and  let  (cci,  y^i  ^-^  be  the  given  point.     Let  the  plane 

X  cos  a  4" ,'/  cos  y8  +  2;  cos  y  =^:»',  (2) 

which  is  evidently  parallel  to  the  given  plane,  pass  through 
the  given  point  (a^i,  ?/i,  z^  ;  then  we  have 

a-i  cos  a  +  2/i  cos  ^-\-z^  cos  y  =  js'.  (3) 

Hence,  x^  cos  a  +  yi  cos  ^^z-^  cos  y  — p  ^=p^  — p. 

But  ^9' — ^9  equals  numerically  the  distance  between  the 
planes  (1)  and  (2),  and  is  therefore  the  required  distance. 
Hence,  to  find  the  distance  of  any  point  from  the  plane 
x  cos  a  +  y  cos  ^  +  «  cos  y =_p, 
substitute  the  coordinates    of  the   point  for  x,   y,  z  in   the 
expression 

X  cos  a-\-  y  cos  /3  +  2:  cos  y  — p. 
CoR.  If  the  equation  of  the  plane  is  Ax-\-  By-\-  Cz=^D,  and 
d  denotes  the  distance  of  (xi,  y^,  z-^)  from  this  plane,  we  have 
Ja-i  +  By^  -\-Cz,  —  D 

The  distance  as  given  by  the  formulas  will  evidently  be 
pjositive  or  negative,  according  as  the  point  and  origin  are  on 
opposite  sides  of  the  plane,  or  on  the  same  side.  The  sign 
maybe  neglected  if  simply  the  numerical  distance  is  required. 

Exercise  44. 

1.  To  which  coordinate  plane  is  3y — 4,?  =  2  perpendicular? 
X  —  8-?  —  7  =  0?  X  —  2y  =  2?  x  =  mz-\-p?  y^nx-\-q? 
What  is  the  locus  of  .-  =  5?  y  =  —  7?  y  =  4:?  z  =  —  2? 
x  =  0?   y  =  0?   z  =  0  ? 


THE    PLANE.  241 

2.  Reduce  to  the  normal  form 

3x  — 2y  +  ;2  =  2;     5iK  —  4 // +  .- =  4. 
Wliat  is  the  distance  of  each  of  these  planes  from  the 
origin?    What  are  tlie  direction  cosines  of  the  perpendicu- 
lars to  each  ?    Wliich  of  the  eight  octants  does  each  truncate? 

3.  Find  the  intercepts  on  the  axes  of  3x  —  2?/  +  4,^  — 12 
=  0;  of  6x  —  4.>/  —  3z  +  24:  =  0;  of  5a;  +  7?/ +  S.v  +  35  =  0. 
Which  of  the  eight  octants  does  each  truncate?  Reduce 
each  equation  to  the  symmetrical  form. 

4.  What  is  the  equation  of  the  plane  at  the  distance  7 
from  the  origin,  and  perpendicular  to  the  line  whose  direc- 
tion cosines  are  proportional  to  2,  —  3,  and  V3  ? 

5.  What  is  the  equation  of  the  plane  whose  intercojjts 
on  the  axes  are  respectively  4, — 3,  — 7?  — 1,  — 2,  — 5? 

6.  Find  the  equation  of  the  plane  passing  through  the 
points  (1,  2,  3),  (0,  4,  —1),  and  (1,  —1,  0). 

7.  Find  the  angle  between  the  planes 

2x^z-,j  =  3, 
z-\-x-{-2i/  =  5. 

8.  Find  the  angle  between  the  planes 

3z  +  5x  —  7>/  =  —  l, 
3z  —  2x—    y  =  0. 

9.  Find  the  angle  that  the  plane  Ax  +  Bij  -\-  Cz  =  D 
makes  with  each  of  the  coordinate  planes. 

10.  Find  the  distance  from  (2,  —3,  0)  to  the  plane 

V3;?  +  2a;  — 3^  =  4. 

11.  Show  that  the  two  points  (1,  — 1,3)  and  (3, 3,  3)  are 
on  opposite  sides  of,  and  equidistant  from,  the  plane 

5x  +  2>j  —  lz^Si^0. 


242  ANALYTIC    GEOMETRY. 

12.  If,  in  Fig.  92,  OM=a,  OE  =  b,  OS=r,  find  the 
equation  of  the  phine  through  the  points  31,  1',  A*.  Find 
the  length  of  the  perpendicular  from  S  ujwu  thi.s  plane. 

13.  Prove  that  the  plane 

A(x  -  X,)  +  B(i/  -  y,)  +  Ciz  -  ,.0  =  0 
passes  through  the  point  (a-i,  t/^,  z-^,  and  is  parallel  to  the 
plane  Ax  -f-  By  -\-  Cz  =  D. 

14.  Find  the  equation  of  the  plane  passing  through  the 
point  (3,  4,  — 1),  and  parallel  to  the  plane  2x  +  4y  —  ,^  =  2. 

15.  What  three  equations  must  be  satisfied  in  order  that 
the  plane  Ax  -\-  By  -\-  Cz  =  D  may  pass  through  the  two 
points  (xi,  yi,  z-^),  (x^,  y^,  z^),  and  be  perpendicular  to  the  plane 

A'x  +  B'y-{-C'z  =  I>'? 

16.  Find  the  equation  of  tlie  plane  passing  througli  the 
points  (1,  1,  1),  (2,  0,  —  1),  and  perpendicular  to  the  plane 
x-\-y  —  z  =  3. 

17.  What  three  equations  must  be  satisfied  in  order 
that  the  plane  Ax  -f-  By  +  Cz  =  D  may  pass  through  the 
three  points  (x^,  v/i,  z^),  (x.,  y^,  z^),  (x^,  y^,  Zs)? 

18.  Find  the  equation  of  the  plane  that  passes  through 
the  points  (1,  2,  3),  (3,  2, 1),  (2,  3, 1),  and  find  the  distance 
of  this  plane  from  the  origin. 

19.  Find  the  equation  of  the  jdane  through  (2,  3,  — 1) 
parallel  to  the  plane  3x  —  4//  -|-  7z  =  0. 

20.  Find  the  equation  of  the  plane  that  passes  through 
the  point  (1,2,3),  and  is  perpendicular .  to  each  of  the 
planes  a;  +  2*  =  1,  y -\-  5z  =  1. 


CHAPTER    III. 


THE    STRAIGHT   LINE. 

226.    To  find  the  equations  of  a  straight  line. 

The  coordinates  of  any  point  on  the  line  of  intersection 
of  two  planes  will  satisfy  the  equation  of  each  of  these 
planes.  Hence,  any  two  simultaneous  equations  of  the  first 
degree  in  a-,  y,  and  ,-.-  represent  some  straight  line.     Of  the 


Fig.  99. 

indefinite  number  of  pairs  of  planes  that  intersect  in,  and 
therefore  determine,  a  straight  line,  the  ecpiations  of  its 
projecting  planes  on  the  coordinate  i)lanes  are  the  simplest, 
and  two  of  them  are  taken  as  the  etpiations  of  the  line. 
Thus,  let  FPl'21l  and  W US  be  the  projecting  planes  of  any 


244  ANALYTIC    GEOMETRY. 

straight  line  PP'  on  the  coordinate  planes  xz  and  yz,  respec- 
tively; and  let  the  equations  of  these  projecting  planes  be 
X  =  mz  -\-p,  (1) 

y  =  nz-\-q;  (2) 

then  are  (1)  and  (2)  the  equations  of  the  line  PP'. 

Cor.  1.  Let  RE  and  SH  be  the  projections  of  the  line 
PP'  on  the  planes  xz  and  yz,  respectively.  Since  the  line 
RE  lies  in  the  plane  PP'RE,  equation  (1)  expresses  the 
relation  between  the  coordinates  x,  z  of  every  point  in  RE  \ 
hence,  (1)  is  the  equation  of  RE  referred  to  the  axes  ZZ' 
and  OX.  For  like  reason,  (2)  is  the  equation  of  the  pro- 
jection SH  referred  to  the  axes  ZZ'  and  OY. 

Hence,         rti  =  tan  ZA  E  =  slope  RE ; 

■p  =  OP  =  intercept  of  RE  on  0X\ 

TO  =  tan  ZBff^  slo^e  SH; 

q  =  0S=^  intercept  of  SH  on  0  Y. 

Eem.  The  locus  of  (1)  in  space  is  the  plane  PP'ER, 
while  its  locus  in  the  plane  xz  is  the  line  RE.  Similarly, 
the  locus  of  (2)  in  space  is  the  plane  PP'HS,  while  its 
plane  locus  is  SH.  The  locus  in  space  of  (1)  and  (2),  con- 
si9.ered  as  simultaneous,  is  the  line  PP'. 

CoR.  2.    Eliminating  z  between  (1)  and  (2),  we  obtain 

n         (         np 

y=i  —  x-\-\  q 

•^       m         \^        VI 

whose  locus  in  space  is  the  projecting  plane  of  PP'  on  the 
plane  xy,  and  whose  locus  in  the  plane  xy  is  the  projection 
of  PP'  on  that  plane. 

CoR.  3.    Making  ?,'  =  0  in  equations  (1)  and  (2),  we  obtain 
x=p,     y=q\ 
hence,   the   line  PP'  pierces    the   plane  xy  in   the  point 
(/>,  q,  0).     This  is  evident  also  from  the  figure. 


THE    STRAIGHT    LINK.  245 

In  like  manner,  we  find  tliat  the  line  })ierce.s  the  planes 
xz  and  yz  respectively  in  the  points 

np  —  mq  0,  _  1 Y     ({),  ^g  ~  '^P^  _  21 
n  ^^/       \  ^  ^"' 

227.  To  find  the  symrtietrical  equatioiis  of  a  rujltt  line. 
Let  a,  (3,  y  be  the  direction  angles  of  any  right  line, 

(^15  l/u  ^i)  some  fixed  point  in  it,  and  (x,  y,  z)  any  other 
point  of  the  line.  Let  r  denote  the  distance  between  these 
two  points.     Then  by  §  218,  Cor.  2,  we  have 

X  —  Xi=^r  cos  a,     y  —  yi  =  r  cos  (3,     z  —  s^  ^  ?■  cos  y.      (1) 

TTtn  2C  —  Xl         y—y\         Z  —  Zl  rn.,-, 

Whence,      -  =  '^ — ^  = >  rCol 

cos  a         COSp        COSY  "■       -* 

which  are  the  symmetrical  equations  of  a  right  line  pass- 
ing through  the  point  (xi,  y^,  z^. 

Cor.    If  [63]  passes  through  a  second  point  (x.^,  y,,  '-'2); 
its  coordinates  must  satisfy  [63] ;  hence,  we  have 

^2  ^1 1/2         Vl         ~2  ^l  ,n-. 

cos  a  cos  /?  cos  y  ^   ^ 

Dividing  each  member  of   [63]   by  the   corresponding 
member  of  (2),  we  obtain 

xz~x\     2/2-2/1     zt—zx  L     -• 

which  ai"e  the  equations  of  a  right  line  through  the  two 
points  (xi,  ?/i,  Si)  and  (a-g,  y^i  z^). 

228.  If  we  divide  the  denominators  in  any  equations  of 
the  form 

x  —  xi^y  —  yi^z  —  zi 
L  M  N  ^^ 


by  \l L^-\-  ]\P-{-  N'^,  the  denominators  will  then  be  the 
direction  cosines  of  some  line  (§  213,  Cur.),  and  the  equa- 
tions will  be  in  the  form  of  [Qo^- 


246  ANALYTIC    GEOMETKY. 

Hence,  to  rcdvice  equations  in  the  form  of  (1)  to  the  sym- 
metrical form,  divide  each  denominator  hij  the  square  root  of 
the  sutn  of  the  squares  of  the  denominators. 

Cor.  The  locus  of  equations  (1)  is  the  line  through 
(xi,7ji,z{)  parallel  to  the  radius  vector  of  the  point  (L,  M,  JV). 

229.    To  find  the  diKjle  between  the  lines 

L     ^     M    ^    N    ' 
and —  ^ — ^-  — -^■ 


L'           M'           N' 
By  §  228  the  direction  cosines  of  these  lines  are  respectively 
L  M  N 

L'  M'  N' 


^L'-'-^-lI'^+N""     \/L"'-\-M"'-\-N'^     \/Z'2+J/'2+^'2 

Substituting  these  values  in  [57],  we  obtain 

„     LL'+  MM'+  NN<  ^^^^ 

cos  e  =    ,  --'  , [651 

Cor.  1.    If  the  lines  are  parallel,  —=-—=—,   and  con- 
versely. 

Cor.  2.    If  the  lines  are  perpendicular, 
LL'  -f  MM<  +  NN'  =  0, 
and  conversely. 

230.    To  find  the  inclination  of  the  line 
x  —  Xi^y  —  ih^z  —  Zy_ 

L  M  N  ^  > 

to  the  plane     Ax  -f-  Bij  -j-  Cz  =^  D.  (2) 


THE    STRAIGHT    LINE,  247 

The  equation  of  the  perpeiidiciilar  from  (x^,  iji,  z-^)  to  the 
plane  IS  a:  —  a^i  _ y  —  ?/i  _z  —  z^ 

A     ~     B     ~     C    '  ^'^^ 

Now,  the  inclination  of  line  (1)  to  plane  (2)  is  evidently 
the  complement  of  the  angle  between  the  lines  (1)  and  (3). 
Denote  this  inclination  by  v;  then  sin  t>  =  cos  ^,  0  being 
the  angle  between  the  lines  (1)  and  (3).     Hence, 

AL  +  BM+  C]>f^  ^,.^^ 

CoR.  1.  If  the  line  is  parallel  to  the  plane,  sin  v  ^  0, 
and,  therefore,  AL-\-  BM-\-  C'i\^=0,  and  conversely. 

Cob.  2.  If  the  line  is  perpendicular  to  the  plane,  sin  v  =:  1, 
and,  therefore,  ^  =  f=f,  and  conversely. 

Cor.  3.    If  line  (1)  lies  in  plane  (2),  then 
AL-{-BM-\-CN=Q, 
and  Ax^  -\-  By^  +  Cz^  =  D, 

and  conversely. 

Exercise  45. 

1.  Determine  the  position,  direction  cosines,  and  direc- 
tion angles  of  the  intersection  of  the  planes  x-{-  y  —  z  + 1 
=  0,  and4a;  +  y/  — 2,~  +  2  =  0. 

Eliminating  successively  y  and  z  between  the  equations, 

X      1/      z  —  1 
we  obtain  ox  —  .~+l  =0  and  2x — '/  =  0:  or  7  =  !.  =  — r; — 

1      -J  o 

From  the  last  form  we  know  that  the  line  passes  through 
the  point  (0,  0, 1),  and  is  parallel  to  the  radius  vector  of  the 
point  (1,  2,  3).  The  direction  cosines  are  found  by  divid- 
ing the  denominators  1,  2,  3,  by  Vl4;  and  the  direction 
ancrles  are  found  from  their  cosines. 


248  ANALYTIC    GEOMETRY. 

2.  Determine  the  position  and  direction  cosines  of  the 
intersection  of  x  —  2y  =  5  and  3x  +  ?/  —  7z  =  0. 

Here  — - —  =  '-  =  "       ^  ,  whence  the  line  passes  through 

the  point  (5,  0,  -y),  and  is  parallel  to  the  radius  vector  of 
(2,  1,  1). 

3.  Determine  the  position  of  the  line 

5a;  —  4?/  =  1,  '6y  —  bz  =  2. 

4.  Wliat  is  the  position  of  the  line  a;  =  3,  2/^=4?     Of 
the  line  y  =  4:,  z  =  —  5?     Of  the  line  x  =  —  2,  z  =  3? 

5.  Find  the  equations  of  the  right  line  passing  through 
the  points  (1,  2,  3),  (3,  4,  1). 

6.  Find  the  points    in  which  the  line  of   Example  5 
pierces  the  coordinate  planes. 

7.  Two  of  the  projecting  planes  of  a  line  are  x-\-?/  =  'i 
and  2x  —  5z  =  —  2 ;  find  the  third. 

8.  A  line  passes  through  (2,  1,  —1)  and  (—3,  —1,  1); 
find  the  equations  of  its  projections  on  the  coordinate  planes. 

9.  Show  that  the  lines  t=9=t  and  3-=^^=  :|^  are 
at  right  angles. 

10.  Show  that  the  line4a;  =  3y  =  —  z  is  perpendicular 
to  the  line  3x=  —  ?/  =  —  4«. 

11.  Find  the  angle  between  the  lines 

X ?/ z  X ?/    z 

12.  Find  the  angle  between  the  right  lines 

y:=OX-\-3,    Z:=3x-\-5, 

and  y^=2x,  z  =  x-\-l. 

13.  Find  the  angle  between  the  lines 

y  =  2.r  +  2,  z  =  2x  +  l, 
and  y^4:X-\-l,  z=   x-\-5. 


THE    STRAIGHT    LINE.  249 

14.  Show  that  the  lines 

3x  +  2i/-{-z  —  5  =  0,x  +  y—2z  —  'S  =  0, 
and  8x  — 4i/  —  4:z  =  0,7x-\- 10//  — 8z  =  0 

are  at  right  angles. 

15.  Find  the  equations  of  the  line  through  ( — 2,  3,  — 1) 
parallel  to  the  line  y=^  —  2x  + 1,  z  =  3x  —  4:. 

16.  Find  the  equations  of  the  line  through  (3,  — 7,  — 5), 
its  direction  cosines  being  proportional  to  —  3,  o,  —  6. 

17.  Find  the  equations  of  the  line  through  (2,  — 4,  — 6) 
perpendicular  to  the  plane  3x  —  6i/-\-2z  =  4:. 

18.  Find  the  inclination  of  the  line 

x  —  4 ?/  +  2 z  —  5 

3     ~  — 2   ~  — 4 
to  the  plane  2x  —  Ay-\-3z^=l. 

19.  Reduce  the  equations,  x  =  mz  -\-p,  y=  nz  -\-  q,  to  the 
symmetrical  form,  and  thus  find  the  direction  cosines  in 
terms  of  m  and  ?i. 

20.  Show  that  the  formula  for  the  angle  included  between 
the  lines 

x  =  mz  +p,   y  =  nz  -\- q, 

and  2"=  vi'x  -\-p',  y  =  7i'z  -\-  q' 

mm'  -\-  nn'  +  1 

IS  cos  6  =  —  = —  ■  • 

Vm'+?i2_^l  Vm'^  +  ?i'2  +  l 

21.  Prove  that  the  lines  in  Example  20  are  perpendicular 
if  vim'  -\- nn'  -\- 1  =  0,  and  conversely.  Prove  that  they  are 
parallel  if  vi  =  m',    and  n  =  n',  and  conversely. 

22.  Prove  that  two  lines  are  parallel  if  their  projections 
are  parallel,  and  conversely. 


250  ANALYTIC    GEOMETRY. 

SUri'LEMENTAllY   PROrOSlTlONS. 

231.  The  Traces  of  a  plane  are  its  lines  of  intersection 
with  the  coordinate  planes.  Thus  AB,  JW,  CA  (Fig.  98) 
are  the  traces  of  the  plane  ABC. 

232.  To  find  the  equations  of  the  traces  of  the  ylane 

Ax-\-By+Cz  =  D.  (1) 

For  every  point  in  the  plane  xy,  z^O  ;  hence,  putting 
«  =  0  in  (1),  we  obtain 

Ax^Bu  =  D,  (2) 

whicli   is  tlie  equation  of  the  trace  AB  (Fig.  98)  on  the 
plane  xy.     For  like  reason, 

By-^Cz=D  -                    (3) 

and               Ax  +  Cz  =  D  (4) 

are  the  equations  of  the  traces  BC  and  CA  on  the  planes 
yz  and  xz,  respectively. 

CoR.  The  perpendicular  from  the  origin  to  (1)  is 

x // z 

A~B^C' 
and  its  projections  on  the  coordinate  planes  are 

Bx  =  Ay,  Cy=Bz,  Cx  =  Az.  (5) 

By  comparing  coefficients,  we  see  that  lines  (5)  are  per- 
pendicular to  (2),  (3),  and  (4),  respectively.     Hence, 

If  a  line  in  sjxice  is  2)(ir2Jendicular  to  a  2)lane,  its  projections 
are  jierpendicular  to  the  traces  of  the  plane. 

233.  To  find  the  conditioii  that  the  right  lines 

X  =  niz  +;y,  y  —  nz  +  q, 

and  X  =  m'z  +  /'',  y  =  n'z  -\-  q', 

may  intersect,  and  to  find  their  points  of  intersection. 


THE    STRAIGHT    LINE.  251 

Equating  the  two  values  of  x  and  y,  we  have 

jj'  — p  q' —  q 

z  = -)  z=- ■• 

7ft  —  nv  n  —  71 

If  the  two  lines  intersect,  these  two  values  of  z  must  be 

p'  —  /'         7'  —  Q  •   • 

equal ;  hence, ^  = -.  is  the  equation  of  condition 

m  —  111       n  —  7i' 

that  the  two   given  lines  in  si)ace  intersect. 

When  this  condition  is  fultilled,  the  values  of  x  and  /j 

may  be  found  by   substituting  either  value  of   z  in  the 

equations  of  either  line. 

234.  To  pass  a  plane  through  the  point  (.Tj,  ?/o,  z^)  and 
the  right  line 

L     ~    31    ~     N    '  ^"^ 

If  the  plane  Ax  +  By  +  Cz  =  B  (1) 

passes  through  the  point  (.Tg,  ?/„,  .-o),  we  have 

Ax^^By.,^Cz.  =  D;  (2) 

and  if  line  (a)  lies  in  plane  (1),  we  have 

Ax^^By,-^Cz,=D,  (3) 

and  AL^BM\CN=^.  (4) 

The  equation  of  the  required  plane  is  found  by  eliminat- 
ing A,  B,  C,  D  from  (1),  (2),  (3),  and  (4). 

To  simplify  the  process  of  elimination,  (1)  might  be  written 
in  the  form  A'x-\-  B'y-\-  C'z  =  1,  but  the  solution  would  be 
less  general,  as  it  would  not  embrace  the  case  when  Z>  ^  0. 

235.  From  the  forms  x  =  mz  -{-j),  y  =  nz-\-q,  show  that 
the  equations  of  a  line  passing  through  (xi,  y^,  z^  are 

\  .r  —  .r,  ==  m  (z  —  z^), 


CHAPTER  IV. 
SURFACES  OF  REVOLUTION. 

236.  It  has  been  shown  that  a  single  equation  of  the 
first  degree  between  three  variables  represents  a  plane 
surface,  and  that  two  such  equations  in  general  represent  a 
right  line.  It  is  evident,  moreover,  that  in  general  three 
such  equations  determine  a  point  common  to  their  loci. 
Thus,  if  in  Fig.  92,  OM=a.,  OE  =  b,  and  OS=c,  then  the 
equations  x  =  a,  y=^h,  z^c  determine  the  point  P,  and  are 
called  its  equations.     We  proceed  to  show  that. 

In  general,  any  single  equation  of  the  form  f  (x.  y,  s  )  =  0 
represents  a  surface  of  some  kind;  two  such  equations  represent 
a  curve,  and  three  determine  one  or  more  pohits. 

(i)  Let  two  of  the  variables  be  absent ;  for  example,  let 
the  equations  be  /  (x)  =  0.  Now  /  (a;)  =  0  may  be  written 
in  the  form 

{x  —  ai)  (a-  —  a^)  {x  —  a^)  (x  —  «.„)  =0,  (1) 

in  which  a^,  Oo,  a^,  «„  are  the  n  roots  of/(a;)=;0.    The 

locus  of  (1)  is  evidently  the  n  parallel  planes  a?  =  «!,  a;  =  a^, 
,  x=-a^.  Similarly,  the  equations/ (?/)  =0,/(s)=:0  repre- 
sent planes  perpendicular  to  the  axes  of  y  and  z,  respectively, 
(ii)  Let  one  of  the  variables  be  absent ;  for  example,  let 
the  equation  be  /  {x,  y)  =  0.  The  locus  of  /(re,  y')=0  in  the 
plane  xy  is  some  plane  curve.  Through  P,  any  point  in  this 
curve,  conceive  a  line  parallel  to  the  axis  of  z  ;  then  the 
coordinates  x,  y  of  all  points  in  this  line  will  equal  those  of 
P,  and  hence  satisfy  the  equation  /  (x,  y)  =  0.     Hence,  the 


SURFACES  OF  REVOLUTION.  253 

locus  in  space  of  /  (x,  y)  =  0  is  the  surface  generated  by  a 
right  line  which  is  always  parallel  to  the  axis  of  z,  and 
which  moves  along  the  plane  locus  oif(x,  y)  =0. 

That  is,  the  locus  in  space  of  f  (x,  y)  =  Q  is  a  cylindrical 
surface  whose  elements  are  parallel  to  the  axis  of  z,  and  whose 
directrix  is  the  pjlane  locus  off  (x,  y)  =  0. 

Similarly,  the  equations  /  (x,  z)  =  0  and  /  (y,  z)  =  0  rep- 
resent cylindrical  surfaces  whose  elements  are  parallel  to 
the  axes  of  y  and  x,  respectively. 

(iii)  Let  the  equation  be  /  (x,  y,  z)  =  0.  If  in  this  equa- 
tion we  putx  =  a  and  y=^b,  the  roots  of  the  resulting 
equation  in  z  will  give  the  points  in  the  locus  that  lie  on 
the  line  through  (a,  b,  0)  parallel  to  the  axis  of  z.  But  as 
the  number  of  these  roots  is  finite,  the  number  of  points 
of  the  locus  on  this  line  is  finite.  Hence,  the  locus  which 
embraces  all  such  points  for  different  values  of  a  and  b 
must  be  a  surface  and  not  a  solid. 

(iv)  Two  equations  considered  as  simultaneous  are  satis- 
fied by  the  coordinates  of  all  the  points  of  intersection  of 
their  loci ;  that  is,  they  represent  the  curve  of  intersection 
of  two  surfaces. 

(v)  Three  independent  simultaneous  eqiiations  are  satis- 
tied  only  by  the  coordinates  of  the  points  in  which  the  curve 
represented  by  two  of  them  cuts  the  surface  represented  by 
the  third ;  hence  they  determine  these  points. 

Cor.  1.  From  (ii)  it  follows  that  x"^ -\- if  =  i^  is  the  equa- 
tion of  a  cylinder  whose  axis  is  the  axis  of  z,  and  whose 
radius  is  r.     Also, 

f  =  ^p:c,~^-  =  l,    and  -.-^=1 

are  equations  of  cylindrical  surfaces  whose  elements  are 
parallel  to  the  axis  of  ,~,  and  whose  directrices  are  respec- 
tively the  parabola,  the  ellipse,  and  the  hyperbola. 


254  ANALYTIC    GEOMETRY. 

CoK.  2.  I (■  F  (x,  i/)=0  is  the  equation  obtained  by 
eliminating  z  between  the  two  equations  /  (a-, //,  s)  =  0  and 
/i  ('^'j  l/>  ^)  =  ^j  then  the  locus  in  space  of  F  (.r,  y)  =  0 
is  the  projecting  cylinder  on  the  plane  xi/  of  the  curve 
represented  by  the  two  equations.  The  j/lane  locus  of 
F(_x,  y):=0  is  the  projection  of  this  curve  on  tlie  plane  xi/. 
If  the  curve  is  parallel  to  the  plane  of  jn-ojection,  the  curve 
and  its  projection  are  equal. 

The  equation  obtained  by  eliminating  x  ov  y  between  the 
two  equations  has  evidently  a  like  interpretation. 

237.  The  Traces  of  a  Surface  are  its  intersections  with 
the  coordinate  planes. 

If/(.^;,  11,  0)=0  denotes  the  equation  obtained  by  mak- 
ing z  =  0  in  f(x,  y,  z)  =  0,  then  the  plane  locus  of  /  (x,  y,  0) 
=  0  is  evidently  tlie  trace  of  the  surface  f  (x,  y,  z)  =0  on 
the  plane  xy. 

Surfaces  of  Revolution. 

238.  A  Surface  of  Revolution  is  a  surface  that  may  be  gen- 
erated by  a  curve  revolving  al)out  a  fixed  straight  line  as  an 
axis.  The  revolving  curve  is  called  the  Generatrix ;  and 
the  fixed  right  line,  the  Axis  of  Revolution,  or  simply  the 
Axis,  A  section  of  the  surface  made  by  a  plane  passing 
through  the  axis  is  called  a  Meridian  Section.  From  these 
definitions,  it  follows  that 

(i)  Every  section  made  by  a  plane  perpendicular  to  the 
axis  is  a  circle,  whose  centre  is  in  the  axis. 

(ii)  Any  meridian  section  is  equal  to  the  generatrix. 

239.  To  find  the  general  equation  of  a  surface  of  revolution. 
Let  the  axis  of  z  be  the  axis  of  revolution,  and  let  P  be  any 

point  in  the  meridian  section  made  by  the  plane  xz.  Let  PHR 
be  a  section  through  P  jterpendicular  to  the  axis  of  z,  and 
denote  the  radius  CH,  or  CP,  of  this  circular  section  by  r. 


SURFACES  OF  UEVOLUTION. 


25/ 


Now  for  all  points  in  this  circular  section,  we  have  »•-  + 
y2=r  /-^^  and  z  =  MP.  The  value  of  r^,  in  terms  of  •:,  is  ob- 
tained by  substituting  r  for  cc  in  the  equation  of  the  me- 
ridian section  made  by  the  plane  zx.  Denoting  this  value 
of  r*  \iy  f(z),  and  equating  tlie  two  values  of  r^,  we  have 

Qc'^+y^  =  f{z),  [G7] 


Fig.  100. 

which  expresses  the  relation  between  the  coordinates  cr,  y,  z 
of  all  points  in  the  section  PHB.  But  as  P  is  any  point 
in  the  meridian  section  NP,  [67]  is  the  general  equation 
of  a  surface  of  revolution  whose  axis  is  the  axis  of  z. 

240.  Paraboloid  of  Revolution.  A  Poraholnid  of  Perolu- 
tion  is  a  surface  that  may  be  generated  by  a  parabola 
revolving  about  its  axis. 

In  this  case  the  equation  of  the  meridian  section  in  the 
plane  zx  is  a;^  =  4/>;s  ; 


hence, 


^^  =  4.pz=f(z). 


256  ANALYTIC    GEOMETRY. 

Substituting  in  [07],  we  obtain 

X"-  +  !/-  =  4rpz,  [68] 

which  is  the  equation  of  the  j^ctt'cil^oloid  of  revolution. 
If  in  [08]  we  put  x  =  m,  we  obtain 

which  is  the  equation  of  the  projection,  on  the  plane  yz,  of 
the  section  of  the  paraboloid  made  by  a  plane  parallel  to 
the  plane  yz,  and  at  a  distance  from  it  equal  to  m.  Now 
the  plane  locus  of  (1),  for  all  values  of  in,  is  a  parabola  ; 
hence,  every  plane  section  of  the  paraboloid  parallel  to  the 
plane  yz  is  a  parabola.  If  in  [68]  we  put  y^n,we  obtain 
x^  =  4pz  —  n^.  (2) 

From  (2)  we  learn  that  all  plane  sections  parallel  to  the 
plane  xz  are  also  parabolas.  From  definition,  we  know 
that  all  plane  sections  parallel  to  the  plane  xy  are  circles. 

241.  Ellipsoid  of  Revolution.  Xn  Ellipsoid  of  Revolu- 
tion, or  Spheroid,  is  a  surface  tliat  may  be  generated  by  an 
ellipse  revolving  about  one  of  its  axes.  It  is  called  Oblate 
when  the  revolution  is  about  the  minor  axis;  and  Prolate 
when  about  the  major  axis. 

(i)  "When  the  revolution  is  about  the  minor  axis,  the 
equation  of  the  meridian  section  in  tlie  plane  xz  is 

7^2  +  7i=l5  lience,  r-'^aH  l--^    =/(^)- 


Substituting  in  [07],  and  reducing,  we  obtain 

S  +  S  +  g=l.  [69] 

which  is  the  equation  of  the  ohlate  spheroid. 
Cor.  1.    If  a  =  h,  [69]  becomes 

x'  +  lf  +  ^^a:^,  (1) 

which  is  the  equation  of  a  sphere  whose  radius  is  a. 


SURFACES  OF  REVOLUTION.  257 

Cor.  2.    If  in  [69]  we  put  x^=in,  we  obtain 

^4---!-  — .  (2-) 

Since  (2)  represents  an  ellipse,  a  point,  or  no  locus  in  the 
plane  yz,  according  as  w?  <C,  =,  or  >  a^,  the  surface  lies 
between  the  two  tangent  planes  a;  =  «  and  a;  =  —  a,  and  all 
plane  sections  parallel  to  the  plane  yz  are  ellipses. 
.    If  in  [69]  we  put  y  =  n,  we  obtain 

Since  (3)  represents  an  ellipse,  a  point,  or  no  locus  in  the 
plane  «cc,  according  as  v?  <,  =,  or  !>  a^,  the  surface  lies 
between  the  two  tangent  planes  y^a  and  y  =  —  a,  and  all 
plane  sections  parallel  to  the  plane  zx  are  ellipses. 

If  in  [69]  we  put  z  =  q,  we  obtain 

x-^-\-f  =  a'^(l-^j\  (4) 

Equation  (4)  represents  a  circle,  a  point,  or  no  locus  in 
the  plane  xy,  according  as  ff  <,  =,  or  >  h"^.  Hence,  the  sur- 
face lies  between  the  tangent  planes  z  =  b  and  z^  —  b,  and 
all  plane  sections  parallel  to  the  plane  xy  are  circles. 

(ii)  When  the  revolution  is  about  the  major  axis,  the 
equation  of  the  meridian  section  in  the  plane  xz  is 

^2+1=1;  hence,  ,-=^^(^l-^,^=/<.). 

Substituting  in  [67],  we  obtain 

f5+|.+  f,=  l.  [70] 

which  is  the  equation  of  the  ^jrolate  spheroid. 

If  in  [69]  we  interchange  a  and  b,  we  obtain  [70]. 
Hence,  we  interchange  a  and  b  in  the  discussion  of  [69]. 


258  ANALYTIC    GEOMETRY. 

242.  Hyperboloid  of  Revolution.  An  Htjperholokl  of 
Reiwlutkm  is  a  suriace  that  ina,y  be  generated  by  an  hyper- 
bola revolving  about  one  of  its  axes.  It  consists  of  one  or 
two  nappes,  or  sheets,  according  as  the  hyperbola  revolves 
about  its  conjugate  or  transverse  axis. 

(i)  If  in  [69]  we  substitute  —  V^  for  h"^,  we  obtain 

^+'4-f^=l,  [71] 

a^     a-     b^  L     J 

which  is  the  equation  of  the  hijperholoid  of  one  nappe. 
If  in  [71]  we  put  x^=vi,  we  have 

2  9  9 

whose  plane  locus  is  an  hyperbola  for  all  values  of  m. 
Hence,  all  plane  sections  parallel  to  the  plane  yz  are  hyper- 
bolas. The  transverse  axis  of  any  one  of  these  hyperbolas 
is  evidently  parallel  to  the  axis  of  y  or  z,  according  as  rn^ 
■<  or  >  a^.  If  m'^  =  a-  (1)  becomes 
a 

Hence,  the  sections  of  [71]  made  by  the  planes  a;  =  ±a 
are  each  two  intersecting  riglit  lines. 
If  in  [71]  we  put  .y  =  ?',  we  have 

X       z  ir 

Hence,  all  plane  sections  of  [71]  parallel  to  the  plane  xz 
are  hyperbolas,  whose  transverse  axes  are  parallel  to  the 
axis  of  X  or  z,  according  as  «-<[  or  >  a^-  and  the  sections 
made  by  the  planes  i/=±a  are  eacli  two  intersecting  right 
lines. 

If  in  [71]  we  put  z  =  q,  we  obtain 
x^     y^_         rf 
a^      a^  If 


SURFACES  OF  REVOLUTION.  259 

whose  plaue  locus  is  a  circle  for  all  values  of  «/.  This  cir- 
cle is  smallest  when  y  ^  0.  This  smallest  circle,  which  is 
the  trace  of  the  liyperboloid  on  the  plane  a"//,  is  called  the 
Circle  of  the  Gorge. 

(ii)  If  in  [70]  we  substitute  —V^  for  V^,  we  obtain 

f?+f?-4=-l,  [T2J 

h-      b'      a^  '-     -■ 

which  is  the  equation  of  the  hyperholoid  of  two  nappes. 

The  discussion  of  [72]  for  parallel  plane  sections  is  left 
as  an  exercise  for  the  student. 

243.  The  Centre  of  a  surface  is  a  point  that  bisects  all 
chords  passing  through  it. 

Central  Surfaces  are  such  as  have  a  centre. 

The  ellipsoids  and  hyperboloids  of  revolution  are  central 
surfaces.  For,  from  their  equations,  it  is  evident  that,  if 
{x\  y',  z')  is  a  point  in  any  one  of  these  surfaces,  ( —  x',  —  y', 
—  z')  is  also  a  point  in  the  same  surface.  But  the  chord 
joining  these  two  points  is  bisected  by  the  origin,  which  is, 
therefore,  the  centre  of  the  surface. 

244.  Cone  of  Revolution.  A  Cone  of  Revolution  is  a 
surface  that  may  be  generated  by  a  right  line  revolving 
about  an  axis  which  it  intersects. 

Here  the  equation  of  the  meridian  section  in  plane  xz  is 
z  =  mx  -\-  c ; 

therefore,        ?-^  =  ( j  =f(z). 

Whence,      in-{x-  +  */-)  -  {z-  <■)-  [73] 

is  the  equation  of  the  cone  of  rerolufiun. 


260 


ANALYTIC    GEOMETRY. 


In  this  equation  c  is  the  distance  of  the  vertex  from  the 
origin  and  m  =  tan  XDB. 


Fig.  101. 

If  c  =  0,  [73]  becomes 

im?{oi?  -\-  ]f^  =  z^.  (1) 

From  (1)  it  is  evident  that  the  cone  is  a  central  surface. 
If  in  (1)  we  put  y  =  n,  we  obtain 
z^        3? 

ri?if)i}      IT?         ' 

whose  plane  locus  is  an  hyperbola  for  all  values  of  n.  Hence, 
all  plane  sections  of  the  cone  parallel  to  the  plane  zx  are 
hyperbolas  whose  transverse  axes  are  parallel  to  the  axis  of 
the  cone.  In  lilie  manner,  we  find  that  all  plane  sections 
parallel  to  the  plane  yz  are  hyperbolas.  If  ^  =  0,  s  =  ±  mx, 
whose  locus  is  two  intersecting  right  lines.  Hence,  any 
plane  section  of  a  cone  parallel  to  its  axis  is  an  hyperbola, 
and  any  section  containing  the  axis  is  two  intersecting  right 
lines. 

A  Conic  Section  is  the  section  of  a  cone  made  by  a  plane. 

245.  To  determine  the  nature  of  any  conic  section  that 
is  not  parallel  to  the  axis  of  the  cone,  we  find  the  equation 
of  any  such  section  referred  to  axes  in  its  own  plane. 


SURFACES    OF    REVOLUTION. 


261 


Let  NFJ^  be  any  section  of  the  cone  VB  YN  passing 
through  the  axis  of  y  ;  then  this  section  will  be  perpendic- 
ular to  the  plane  xz.     The  cone,  and  therefore  the  section, 


is  symmetrical  with  respect  to  the  plane  xz.  Refer  this 
section  to  ON  and  0  Y  as  the  axes  of  x  and  y  respectively. 
Let  (cc,  y,  z)  be  the  point  P  referred  to  the  coordinate  planes, 
and  {x\  xj)  be  P  referred  to  ON  and  OY.  Let  XON=  cf> 
and  0DV^=6.  Draw  PJf  perpendicular  to  ON;  then  it 
will  be  perpendicular  to  the  plane  xz,  and  we  have 

y  =  y',  0B=  OM  cos  <l>,  or  a;  =  a:'cos  <f> ; 

BM=  Oil/sin  <^,  or  s^cc'sin  ^. 
Substituting  these  values  of  x,  y,  z  in  [73],  we  obtain 

tan^  6  (cc'2  cos^  <^  +  y'^)  =  (x'  sin  <f>  —  cf. 
Omitting  accents  and  performing  indicated  operations, 
we  have 

?/^tan-^+a'-(cos-</)  tan  -  9  —  sin^<;i))  -\-2cx  sin  ^  —  c-=0. 
Substituting  cos^  </>  tan-  ^  for  sin-  ^,  we  obtain 
2/2  tan2  9  -t-ic^  cos2  <}>  (tan2e  -  tau2  (j>)+  2cac  sin  <}>  -  c-  =  O,  [74] 


262  ANALYTIC    GEOMETRY. 

which  is  the  equation  of  the  conic  ^ViV/ referred  to  ON 
and  0  Y  as  axes. 

By  giving  to  c  all  values  between  0  and  oo  ,  and  to  <^  all 
values  between  0°  and  90°,  equation  [74]  is  made  to  rep- 
resent any  section  of  a  cone  except  those  parallel  to  its  axis, 
which  have  already  been  considered. 

Discussion  of  equation  [74]. 

Here  2=4  cos^  <^  tan^  9  (tan-  0  —  tan-  cf>), 

A= 4c^  [cos-  <j)  tan-  6  (tan-  ^—  tan^  </>)  -f-tan^  6  sin-  ^]. 

(i)   Fh'.sf  su2)j)ose  c  not  equal  to  zero. 

Let  (f><iO  ;  then  tan-  <^  <C  tan'^  6,  2  is  positive,  and  A  is 
not  zero  ;  lience  the  section  is  an  ellipse. 

Let  <^  =  ^ ;  then  tan^  cj>  =  tan^  $,  2  =  0,  and  A  is  not  zero  ; 
hence  the  section  is  a  parabola. 

Let  <ji'>0;  then  is  tan" <^  > tan- ^,  2  is  negative,  and  A 
is  not  zero  ;  hence,  tlie  section  is  an  hyperbola. 

Hence,  when  the  cutting  ])lane  does  not  pass  through  tlie 
vertex  of  the  cone,  the  section  is  an,  ellipse,  a  i>arahola,  or 
an  hyperholn,  according  as  the  angle  ibhicli  the  cutting  plane 
makes  with  the  base  of  the  cone  is  less  than,  equal  to,  or 
greater  than  tlutt  made  hij  an  element. 

(ii)  If  c  =  0,  A  =:  0  ;  henee,  when  the  cutting  plane  passes 
through  the  vertex,  the  elliptical  section  reduces  to  a  point, 
the  parabolic  to  a  straight  line,  and  the  hyperbolic  to  two 
intersecting  right  lines. 

If  </)  =  0,  the  cutting  plane  is  perpendicular  to  the  axis 
of  the  cone,  and  equation  [74]  becomes 

rf  +  .r-  ^  c^  eot^  Q, 
whose  locus  is  a  circle. 

If  c  =  a3,  the  cone  becomes  a  cylinder,  and  the  section 
made  by  a  plane  parallel  to  an  element  is  two  parallel  lines 
or  a  single  rigrht  line. 


SURFACES  OF  REVOLUTION.  263 

Bxercise  46. 

1.  What  is  the  locus  in  space  of  x^  -{-  3x^  —  6a;  —  8  =  0  ? 
of  2/3— 2/  — 5y/  +  6  =  0?    of  s;^  +  m,t;  =  0  ? 

2.  What  is  the  locus  in  space  of  //-=:^Sx?  of  4a;--|-9iy- 
=  36  ?  of  9^2  — 16 //2  z=  144  ?  of  (2a  —  z)  {f  —  h'-)  =  0  ? 
of  z'-\-x-  =  i^? 

3.  Find  the  equations  of  the  projecting  cylinders  of  the 
curve  x"  +  3i/  —  2,^'  =  8,  x'  +  2f  +  3,^^  =  16  ? 

4.  Find  the  equations  of  the  projections  of  the  curves 

^^  +  y^  +  2^~16,  9(.r^  +  /)+4^2  =  36. 

5.  Find  the  semi-axes  and  eccentricity  of  the  ellipse 

4a;2  +  9/  +  4,^2  =  37,  ;i'  =  i. 

6.  Find  the  nature  of  the  curves 

x'^-if^^z'  =  2b,l{x'^>f)-l^  =  m 

7.  Find  the  traces  of  tlie  surface  2x--\-bif  —  7.^'  =  9; 
of  the  surface  x^  -\-  3//'^  =  8z. 

8.  Find  the  equation  of  the  surface  of  revolution  Avhose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  ,-;=  ±  3a'  +  5  ; 
find  its  trace  on  the  plane  xi/. 

9.  Find  the  equation  of  a  cone  of  revolution  one  of 
whose  traces  is  3c^-\-y^  =  9,  and  whose  vertex  is  (0,  0,  5). 

10.  Find  the  equation  of  the  paraboloid  of  revolution  one 
of  whose  traces  is  2x^^^oz  -{-  5. 

11.  Find  the  equation  of  the  paraboloid  of  revolution  one 
of  whose  traces  is  i/'-^=  8x. 

12.  Find  the  equation  of  the  cone  of  revolution  whose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  2^/  ^±.~  +  6  ; 
find  its  vertex. 


264  ANALYTIC    GEOMETRY. 

13.  Find  the  equation  of  the  surface  of  revolution  whose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  ^3^-\-^z'^  =  36. 

14.  Find  the  equation  of  the  surface  of  revolution  whose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  16y^-\-9z'^ 
=  144. 

15.  Find  the  equation  of  the  surface  of  revolution  whose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  dz"^  —  4?/^ 
=  —  36. 

16.  Find  the  equation  of  the  surface  of  revolution  whose 
axis  is  the  axis  of  z,  and  one  of  whose  traces  is  z^x  =  l; 
also  when  one  trace  is  z^  ^=  2if. 

17.  Each  element  of  a  cone  makes  an  angle  of  45°  with 
its  axis  ;  find  the  semi-axes  of  the  section  made  by  a  plane 
cutting  the  axis  5  below  the  vertex  and  at  an  angle  of  60°. 

SUPPLEMENTARY   PROPOSITIONS. 

246.   To  find  the  (/eneral  equation  of  the  sphere. 

Let  r  denote  the  radius  of  any  sphere,  (a,  b,  c)  its  centre, 
and  (x,  y,  z)  any  point  on  its  surface.  Then,  since  r  is  the 
distance  between  the  points  {a,  b,  c)  and  (x,  y,  z),  we  have 
{X  -  a)2  +{y-  6)2  +  (s  -  c)^  =  »-2,  [75] 

or  x'^+f-\-z'  —  2ax  —  2by  —  2cz=^r''—a''  —  b^  —  c\  (1) 
which  is  the  general  rectangular  equation  of  the  sphere. 

If  the  origin  is  at  the  centre,  then  a  =  5  =  c  =  0,  and  [75] 
becomes  ^2  _^  ^2  _^  .2  ^  ^-2^  (2) 

From  (1)  it  follows  that  any  equation  of  the  form 

x''-\-y^  +  z'^Gx  +  Hy  +  Iz  =  K  (3) 

is  the  equation  of  a  sphere. 

Any  equation  of  the  form  of  (3)  can  readily  be  reduced  to 
the  form  of  [75],  from  which  the  centre  and  radius  of  its 
locus  become  known. 


THE    SPHERE.  265 

Since  (3)  or  [75]  contains  four  arbitrary  constants,  a 
sphere  may  in  general  be  made  to  pass  through  any  four 
given  points. 

247.  The  intersection  of  two  spheres  is  a  circle. 

Let  the  equations  of  the  two  spheres  be 

x'+f  +  z'-^Gx  +  Hy  +Iz=K,  (1) 

and  a;2  +  y2  _|_  ^2  ^  q^^  _|_  fj,y  _|_  jr^^  j^,  ^2) 

Subtracting  (2)  from  (1),  we  obtain 

{G  —G')x^  (B  —  H')i/^  (/-  T)  z  =  K—K'.    (3) 

Hence,  the  intersection  of  the  spheres  (1)  and  (2)  lies  in 
the  plane  (3),  and  is  the  same  as  the  intersection  of  (1)  and 
(3).  But  the  plane  section  of  a  sphere  is  a  circle.  Hence, 
the  intersection  of  the  two  spheres  is  a  circle. 

248.  To  find  the  equation  of  the  tanr/ent  plane  to  a  sphere 
at  a  given  point. 

Let  the  given  point  be  (cci,  y^,  z^;  then  the  equation  of 
the  radius  to  this  point,  that  is,  of  the  line  passing  through 
(a,  h,  c)  and  (x^,  y^,  z^)]  is 

x^:^Xi  _ y  —  //i  _z  —  Zi 


a  —  Xi      h  —  y^       G  —  z-^ 

Now  the  tangent  plane  is  perpendicular  to  (1)  at  the 
point  (ccj,  yi,  z-^\  but  the  equation  of  the  plane  through 
(xi,  yx,  Zi)  perpendicular  to  (1)  is 

(a-x,)  (x - X,)  +  (b- y,)  {y -  y,)  +  (c-z,)  (z -  z,)  =0,  (2) 

which  is,  therefore,  t/ie  equation  of  the  tangent  jdane. 

If  the  origin  is  at  the  centre,  a=  ?;  =  c  =0,  and  (2)  becomes 


266 


ANALYTIC    GEOMETRY. 


Transformation  of  Coordinates. 

249.  To  cJiange  the  orirjin  of  coordinates  without  changing 
the  direction  of  the  axes. 

Let  (m,  n,  q)  be  the  new  origin  referred  to  the  old  axes. 
Let  X,  y,  z  be  the  okl,  and  .r',  y\  z'  the  new  coordinates 
of  any  point  F;  then,  evidently,  we  have 

x^m-^  x',  1/  =  n-\-  y',  z  =  q-\-z'. 

Hence,  to  find  the  equation  of  a  locus  referred  to  new 
parallel  axes  whose  origin  is  (vi,  n,  q),  substitute  m-\-x, 
n  +  //,  cmd  q  +  z,  for  x,  y,  and  z,  resjjectively. 

250.  To  change  the  direction  of  the  axes  ivithout  changing 
the  origin. 

Let  ai,  /3i,  yi ;  ag,  /iJg,  72  5  "s,  /^s,  ys  be,  respectively,  tlie  direc- 
tion angles  of  the  new  axes  0X\  0  Y\  OZ'  referred  to  the 


Fig.  103. 


old  axes  OX,  0  Y,  OZ.  Let  x,  y,  z  be  the  old,  and  x',  y\  z'  the 
new  coordinates  of  any  point  P.  Draw  PN  perpendicular 
to  the  plane  X'OY',  and  JVJf  perpendicular  to  OX;  then 
OM^^x',  MN=  y',  and  NP^z'.     Now  the  projection  of 


QUADRICS.  2G7 

OP  on  OX  {  =  ^)  is  equal  to  the  sum  of  the  projections  of 
OM,  MN,  and  NP  on  the  same  line  ;  hence, 

.'C=:x'c0S  ai  +  ^'C0Sa2  +  ;s'C0S  ttj.  (1) 

In  like  manner,  we  obtain 

?/ =  CC' COS  ^1  +  i/' cos  ySo  +  s;' COS  ySs,  (2) 

and  z=^x' cos  yi  +  y'  cos  yo  +  z' cos  yj.  (3) 

Hence,  to  change  the  direction  of  the  axes  witliout 
changing  the  origin,  stibstitiite  for  x,  y,  and  z  their  values 
as  given  in  equations  (1),  (2),  and  (3), 

Since  the  values  of  x,  y,  z  are  each  of  the  first  degree  in 
x',  y\  z\  any  transformation  of  coordinates  cannot  change 
the  degree  of  an  equation.     (§  91.) 

251.  Quadrics.  The  locus  of  an  equation  of  the  second 
degree  that  contains  three  variables  is  called  a  Quadric. 
Thus  the  general  equation  of  a  quadric  is 

Ax'^By'  +  C^  +  Dxy  +  Exz  +  Fyz  -[-Gx+Hy 
-\-Iz  +  K=0.  (1) 

Putting  z  =  q  in  (1),  we  obtain 

Ax^  +  D^r->,  +  /,y  ^(^Eq  +  G)x  +  (Fq  +H)y 
+  (tVr  +  /'/  +  A')-0.  (2) 

Since  the  locus  of  (2)  in  the  plane  xy  is  a  conic,  and  since 
the  coefficients  A,  D,  B  are  tlie  same  for  all  values  of  q,  all 
plane  sections  of  the  quadric  (1),  parallel  to  the  plane  a-y,  are 
similar  conies.  Now  the  axis  of  coordinates  may  be  so 
changed  that  the  new  plane  xy  will  be  one  of  any  system  of 
parallel  planes  cutting  the  quadric.  But,  as  this  transforma- 
tion does  not  change  the  degree  of  the  equation,  it  follows  that 

All  parallel  plane  sections  of  any  quadric  are  similar  conies. 


268  ANALYTIC    GEOMETRY. 

252.  By  transformations  of  coordinates  the  general  equa- 
tion (1)  of  §  251  may  be  reduced  to  one  of  the  two  following 
simple  forms :  * 

Px'-\-Qu^-\-Rz'=S.  (1) 

Px^+Qy'=Uz.  (2) 

Now  whatever  be  the  values  or  signs  of  P,  Q,  R,  S,  equa- 
tion (1)  evidently  represents  central  quadrics.  But  the  loci 
of  (2)  have  no  centre ;  for  if  tliey  had,  and  the  origin  were 
changed  to  that  centre,  the  first  power  of  z  would  disappear 
from  the  equation.  But  no  expression  of  the  form  q -}- z, 
when  substituted  for  z,  can  cause  z  to  disappear. 
Hence,  (2)  represents  non-central  quadrics. 

253.  Central  Quadrics.  If  no  one  of  the  coefficients 
P,   Q,  R,  S  is  zero  in  (1)  of  §  252,  we  have 

S^P^  S^Q^  S-^  R 
which  can  be  written  in  the  form 

*  By  changing  the  direction  of  the  axes  the  general  equation  can  in 
all  cases  be  reduced  to  the  form 

Px2  +  Qy^  +  Rz^  +  G'x  -f  H'y  +  I'z-  K-O.         (1) 
This  transformation  is  analogous  to  that  in  §  189. 
(i)  If  no  one  of  the  three  coefficients  P,  Q,  R  is  zero,  by  a  change 
of  origin,  as  in  §  188,  we  obtain 

Px2  -f  Q?/  +  Rz"-  =  S.  (2) 

(ii)  If  any  one  of  these  coefficients  is  zero,  for  example  R,  by  a 
change  of  origin,  we  obtain 

Px2  -I-  Qy^  =  Uz.  (3) 

If  two  of  these  coefficients  are  zero,  (1)  can  be  reduced  to  a  form 
embraced  in  (3)  by  first  changing  the  origin  and  then  the  direction  of 
the  axes. 


QUADRICS.  269 

5  +  f!-|=l.  (B) 

according  as   S-i-F,  S~Q,  S-^B  are  all  positive,  two 
positive  and  one  negative,  or  one  positive  and  two  negative. 
[If  all  three  are  negative  there  is  no  real  locus.] 
If  S  is  zero,  we  have 

Px'-\-Q!/'  +  Bz'  =  0.  (D) 

If  P,  Q,  or  B  is  zero  in  (1),  its  locus  is  a  cylindrical 
surface  by  (ii)  of  §  236. 

254.  A  discussion  of  (A)  discovers  the  following  proper- 
ties of  its  locus  : 

(i)  Its  traces  on  each  of  the  coordinate  planes  are  ellipses, 
(ii)  All  plane  sections  parallel  to  either  coordinate  plane 
are  similar  ellipses. 

(iii)  The  quadric  is  included  between  the  tangent  planes 

X  =  dz  a,  1/  ^^  ±  b,  z  ^^  ±  c. 

The  quadric  (A)  is  called  an  Ellipsoid.  If  a  =b,  the  ellip- 
soid is  the  oblate  or  prolate  spheroid,  according  as  a  ^or  <C  e. 

The  ellipsoid  may  evidently  be  generated  by  a  variable 
ellipse  moving  parallel  to  the  plane  2:3/  with  its  centre  in  the 
axis  of  z,  its  axes  being  chords  of  the  traces  of  the  quadric 
on  the  planes  yz  and  zx. 

255.  From  a  discussion  of  (B),  we  learn  that : 

(i)  Its  trace  on  the  plane  xij  is  an  ellipse,  while  its  traces 
on  the  planes  ijz  and  zx  are  hyperbolas,  whose  transverse 
axes  lie  on  tlie  axes  of  y  and  x  respectively. 

(ii)  All  plane  sections  parallel  to  the  plane  a-// are  ellipses, 
while  all  plane  sections  parallel  to  the  plane  i/z  and  zx  are 


270  ANALYTIC    GEOMETRY. 

hyperbolas.  The  smallest  elliptical  section  is  the  trace  on 
the  plane  xy.     The  semi-axes  of  this  ellipse  are  a  and  h. 

The  locus  of  (B)  is  called  the  Hyperboloid  of  One  Nappe. 

If  a=^b,  the  locus  of  (B)  is  an  hyperboloid  of  revolution. 

The  hyperboloid  can  evidently  be  traced  by  a  variable 
ellipse  parallel  to  the  plane  xy,  whose  centre  moves  along 
the  axis  of  z,  and  whose  axes  are  the  chords  of  the  traces 
of  the  quadric  on  the  planes  yz  and  zx. 

256.  From  a  discussion  of  (C),  we  learn  that : 

(i)  Its  traces  on  the  planes  yx  and  zx  are  hyperbolas 
whose  tranverse  axes  are  on  the  axis  of  x. 

(ii)  The  plane  sections  parallel  to  the  plane  zy  are 
ellipses,  and  no  portion  of  the  quadric  lies  between  the 
tangent  planes  a;  =  ±  a. 

(iii)  The  plane  sections  parallel  to  the  planes  yx  and  zx 
are  hyperbolas  whose  transverse  axes  are  parallel  to  the 
axis  of  X. 

The  locus  of  (C)  is  called  the  Hyperboloid  of  Two  Nappes. 

257.  If  the  coefficients  of  (D)  are  all  positive  or  all  nega- 
tive, its  locus  is  the  point  (0,  0,  0).  If  two  coefficients  are 
negative  and  one  positive,  by  dividing  by  — 1,  two  become 
positive  and  one  negative.  Hence,  we  need  discuss  only  the 
form  represented  by 

2  2  2 

X  II  Z 

from  which  we  learn  that  : 

(i)  All  plane  sections  parallel  to  the  planes  yz  and  zx  are 
hyperbolas  whose  transverse  axes  are  parallel  to  the  axis  of  z. 

(ii)  All  plane  sections  parallel  to  the  plane  xy  are 
ellipses,  the  trace  on  this  plane  being  a  point. 

(iii)  The  traces  on  the  planes  yz  and  zx  are  each  two 
right  lines  intersecting  at  the  origin. 


QUADRICS.  271 

(iv)  All  plane  sections  throngh  the  axis  of  z  are  two 
right  lines  intersecting  at  the  origin. 

For,  denote  any  plane  through  tlie  axis  of  z  by 

y  =  mx.  (1) 

Eliminating  y  between  (1)  and  {D'),  we  obtain 

z  =  ±^yjb''  +  a'm\  (2) 

ab 

Now  the  intersections  of  (1)  and  (D')  are  the  same  as 
the  intersections  of  (1)  and  (2),  which  are  evidently  two 
right  lines  passing  through  the  origin. 

Hence,  the  locus  of  (D')  is  a  cone  whose  axis  is  the  axis 
of  z,  and  Avhose  directrix  is  an  ellipse.  If  a  =  b,  it  becomes 
a  cone  of  revolution. 

258.  Non-Central  Quadrics.  If  no  one  of  the  coefficients 
P,  Q,  U  is  zero  in  (2)  of  §  252,  we  have 

^^       I       //     ^ 
which  can  be  written  in  the  form 

1-1="'  w 

according  as  P  and  Q  have  like  or  unlike  signs. 

A  discussion  of  (E)  discovers  the  following  properties: 
(i)  Plane  sections  parallel  to  the  plane  xy  are  ellipses, 
and  the  surface  lies  above  the  tangent  plane  ?.'  =  0. 

(ii)  All  plane  sections  parallel  to  the  plane  yz  or  zx  are 
parabolas,  and  the  traces  on  these  planes  are  parabolas, 
having  the  axis  of  z  as  their  common  axis  and  their  con- 
cavities upward. 


272 


ANALYTIC    GEOMETKY. 


By  a  discussion  of  (F)  we  learn  that  : 

(i)  The  traces  on  the  planes  yz  and  zx  are  parabolas 
whose  axes  lie  on  the  axis  of  z,  and  whose  concavities  are 
in  opposite  directions. 

(ii)  Plane  sections  parallel  to  the  planes  yz  and  zx  are 
parabolas  whose  concavities  are  in  opposite  directions. 

(iii)  Plane  sections  parallel  to  the  plane  xy  are  hyper- 
bolas whose  transverse  axes  are  parallel  to  the  axis  of  x,  or 
y,  according  as  z  is  positive  or  negative.  The  trace  on 
this  plane  is  two  intersecting  right  lines. 


DiAGUAMS. 


Note.    These  figures  are  taken,  by  permission,  from  W.  B.  Smith's 
Geometry. 


Elliptic  Paraboloid. 
FA  =  2a  and  GB  =  26  are 
half -parameters . 


Simple  Hyferboloid. 

AB'is,  Ellipse  of  the  Gorge. 
EOT)  is  the  Asymptotic  Cone. 


DIAGRAMS. 


273 


Hyperbolic  Paraboloid. 
OC  and  OD  are  Parabolas. 
OA  and  OB  are  Asymptotic  directions  for  the  Hyperbolas. 


^^ 

C 

,.,,--— 

--"-p-     -^-----^ 

\                 B 

\\}^^ 

Ellipsoid. 
OA  =a,  0B  =  b,  OC  =  c. 


Double  Hyperboloid. 
EOD  is  the  Asymptotic  Cone. 


AISTSWERS. 


Exercise  3.     Page  7. 

1.    Let  xi  =  —  2,  ?/i  =  5,  X2  =  —  8,  7/2  =  —  3.     Substituiing  in  [1], 
we  have 


d  =  V(—  6)2  +  (-  8)2  =  VIOO  =  10. 

In  Fig.  3,  the  points  P  and  Q  are  plotted  to  represent  tliis  case.  If 
we  choose  to  solve  the  question  without  the  aid  of  [1],  we  may  neglect 
algebraic  signs,  and  we  have 

qR  =  NO  —MO  -    8—    2=      6 ; 
PR  =  PM  +  MR  =    b+    Z=      8  ; 
...  Pq2  =^  q]^2  ^  ^2  _  o(5  ^  (54  _  iQQ^  and  Pq  =  10. 


2. 

13. 
5. 

8. 
9. 

5,  5,  6. 

3. 

a,  b,  ■\la:-  +  62. 

4. 

10. 

10. 

V29,  5,  2VT0,  4V5; 

5. 

2  Va2  +  bK 

2VT0,  3VT3. 

6. 

25,  29,  20  V2. 

11. 

8  or  -  16. 

7. 

2Vl7^  6V2,  A 

/1O6. 

12. 

(x  -  7)2  +(y  +  2)2  =  121. 

(X 

-2)2  +  (y-3)2 

=  (x- 

_4)-2 

'■  +  {y- 

-  5)2,  which  reduces  to  x+y—1. 

Ezercise  4.     ] 

Page  9. 

13. 


1.  (6,  6).  3.    (2,  -  2).  5.    (7,   1). 

2.  (-1,  0).  4.   (3,  -1),  (1,  -V),  (-i,  -I).     6.    (a,  -6). 
7.   Take  the  origin  of  coordinates  at  the  intersection  of  the  two 

legs,  and  the  axes  of  x  and  y  in  the  directions  of  the  legs.  Then,  if  a 
and  b  denote  the  lengths  of  the  legs,  the  coordinates  of  the  three  ver- 
tices will  be  (0,  0),  (a,  0),  and  (0,  h). 

10.  Observe  that  now  the  distances  BB  and  BQ  will  be  x  —  x-z  and 

y-y2-  12.  (f,  i).     14.  (7f, -311). 

11.  (6,2).  13.    (8,0).        15.    (13,  -  1),  (-11,  5),  (1,  -  11). 


Is  ANALYTIC    GEOMETRY. 

Exercise  7.  Page  23. 

1.  12,  16.  14.  Locus  does  not  cut  the  axes. 

2.  -  10,  6.  15.  (5,  7). 

3.  ±4,  ±4.  16.  (2,  1). 

4.  ±  I,  ±  2.  17.  (3,  4)  and  (-  4,  .3). 

5.  ±  I,  imaginary.  18.  (3,  4). 

6.  ±  t,  -  4.  19.  (5,  3)  and  (3,  5). 

7.  ±b,  ±  a.  20.  (0,  0)  and  (2,  4). 

8.  3  on  axis  of  x.  21.  (5,  —  3),  (6,  4),  (-  4,  -  1). 

9.  db  3  on  axis  of  x.  22.  V61,  5,  2  V26. 

10.  Locus  passes  througli  origin.  23.  3,  4,  5. 

11.  Locus  passes  through  origin.  i  (a,  b)  (—  a,  6), 

(  On  axis  of  x,  8,  and  —  4.  I  {—  a,  —  b)  [a,  —  b). 

I  On  axis  of  ?/,  4  ±  4  V3.  25.  No. 

-(  On  axis  of  X,  0  and  4,  „«  in 
(  On  axis  of  y,  0  and  8. 

Exercise  9.     Page  31. 

1.  Let  X  and  y  denote  the  variable  coordinates  of  the  moving  point. 
Then  it  is  evident  that  for  all  positions  of  the  point  y  =  3x.  There- 
fore, the  required  equation  is  y  =  3x,  or  y  —  3x  =  0.  Does  the  locus 
of  this  equation  pass  through  the  origin  ? 

2.  X  -  6  =  0,  X  +  6  =  0,  X  =  0. 

3.  y  —  i  =  0,  y  +  \  =  0,  y  =  0. 

4.  Tlie  line  x  =  3  is  tlie  Une  AB  (Fig.  74);  how  is  this  line  drawn  ? 
The  locus  of  the  variable  point  consists  of  the  two  parallels  to  AB, 
drawn  at  the  distance  2  from  AB.  Let  CD,  EF,  be  these  parallels, 
and  (x,  y)  denote  in  general  the  variable  point,  then,  for  all  points  in 
CD,  X  =  3  +  2  =  5,  and  for  all  points  in  EF,  x  =  3  —  2  =  1.  There- 
fore, the  equation  of  the  line  CD  is  x  —  5  =  0,  and  that  of  the  line 
EF  is  X  —  1  =  0.  The  product  of  these  two  equations  is  the  equa- 
tion (x  —  5)  (x  —  1)  =  0.  This  equation  is  evidently  satisfied  by  every 
point  in  each  of  the  lines  CD  and  EF,  and  by  no  other  points.  There- 
fore, the  required  equation  is  (x  —  5)  (x  —  1)  =  0,  or  x^  —  6x  +  5  =  0. 
Verify  that  this  equation  is  satisfied  by  points  taken  at  random  in  the 
lines  CD  and  EF. 


ANSWERS.  3 

5.  2/2  —  10?/  +  16  =  0,  two  parallel  lines. 

6.  x^  +  8x  —  9  =  0,  two  parallellines.         7.  x+  ^^=^0,  y  —  2—0. 
8.    It  is  proved  in  elementary  geometry  that  all  points  equidistant 

from  two  given  points  lie  in  the  perpendicular  erected  at  the  middle 
point  of  the  line  joining  the  two  given  points.  This  perpendicular  is 
the  locus  required,  and  its  equation  evidently  is  x  =  3. 


r 

F 

B 

D 

0 

X 

E 

A 

C 

Y 

P 

\ 

0 

A      X 

Fig.  74. 


Fig.  75. 


Let  US  now  solve  this  problem  by  the  analytic  method.  Let  0 
(Fig.  75)  be  the  origin,  A  the  point  (6,  0),  and  let  P  represent  any  posi- 
tion of  a  point  equidistant  from  0  and  A,  x  and  y  its  two  coordinates. 

Then  from  the  given  condition 

PO  =  PA. 

Therefore,  x^  +  y- =  {x  —  6)2  +  {y  —  0)2, 

or  x2  + 7/2  =  a;2_  12X  +  36-I-2/2; 

whence  x  =  3, 

the  equation  of  the  locus  required. 
9.   X  — 1  =  0.      10.  y -2-0.      11.  x  —  Sy  —  l  =  0.     12.  x  —  y=0. 

13.  x2  +  ?/2  =  100,  a  circle  with  the  origin  for  centre  and  10  for 
radius. 

14.  Express  by  an  equation  the  fact  that  the  distance  from  the 
point  (x,  y)  to  the  point  (4,  —  3)  is  equal  to  5.  The  equation  is 
(x-4)2  +  (y-|-3)2=25. 

15.  (X  +  4)2  +  (y  +  7)2  =  64.  16.   x2  +  i/2  =  81. 

17.    Draw  AO  ±  to  BC  (Fig.  76).     Take  A 0  for  the  axis  of  x,  and 
BC  for  the  axis  of  y ;   then  A  is  the  point  (3,  0). 
Let  P  represent  any  position  of  the  vessel,  x  and  y  its  coordinates 


ANALYTIC    GEOMKTKV. 


OM  and  PM.     Join  PA,  and  draw  PQ  ±  liC,  and  meeting  it  in  Q. 
Then  from  the  given  condition 

PA  -  PQ=  OM. 

Therefore,  ¥1^  -  0M\ 

Now    PA^  =  aJi'^  +  PM'  -  {X-  3)2  +  2/2^   ad  Um'^  =  x2.       Sub- 
stituting, we  have      /^  —  3)2  +  w2=  x^  ■ 
whence  y-  =  6x  —  9. 


Fig.  7G. 

The  locus  is  the  curve  called  the  parabola.     We  leave  the  discus- 
sion of  the  equation  as  an  exercise  for  the  learner. 

18.  If  BC  is  taken  for  the  axis  of  y,  and  the  perpendicular  from  A 
to  BC  as  the  axis  of  x,  the  required  equation  is  2/2  =  i2x  —  3G. 

19.  x2  —  32/2  =  0,  two  straight  lines. 

20.  x2  +  2/2  =  fc2  —  a2,  a  circle. 

21.  4ax  ±  k-  =  0,  two  straight  lines. 

Exercise  10.     Page  33. 

4.  d  =  Vxi2  H-  2/1-'.  6.    X  +  2/  =  7. 

5.  (X  -  4)2  +  (2/  -  6)2  =  64.  7.    (Y,  f ) ;   f  ^^. 

8.  Take  two  sides  of  the  rectangle  for  the  axes,  and  let  a  and  b 
represent  their  lengths  ;  then  the  vertices  of  the  rectangle  will  be  the 
points  (0,  0),  (a,  0),  (a,  ft),  (0,  h). 

9.  Take  one  vertex  as  the  origin,  and  one  side,  a,  as  the  axis  of  x; 
then  (0,  0)  and  (o,  0)  will  be  two  vertices.  Let  (6,  c)  be  a  third  ver- 
tex ;   then  (a  -f-  b,  c)  will  represent  the  fourth. 


I  ANSWERS.  O 

10.    (11,  2),  (-  1,  4),  (15,  16).  11.    (5,  -  2),  (i,  y),  (3,  -  V). 

12.    (1,  -  f).  13.   Vl7.         14.    (5,  V).  16.    (6,  23). 

__     /Xi  +  3x2    2/1  +  Sy2\     /X1+X2     yi±y2\     ( 3Xi+X2    %i-h/2\ 

21.    3  or  —  23,  23.    (8,  6)  and  (8,  -  G). 

^  3  and  2  on  OX  24.    (2  a,  a)  and  (—  2  a,  a). 

I  6  and  1  on  OY.  25.    (a,  0)  and  (-  a,  0). 

26.  10,  2  V26,  2  Vl3. 

27.  Taking  the  fixed  lines  for  axes,  the  equation  is  2/=6x,  or  x=6?/. 

28.  Taking  J.  for  origin,  and  AB  for  the  axis  of  x,  the  equation  is 
x2  -  3?/2  =  0. 

29.  Taking  the  fixed  line  and  the  perpendicular  to  it  from  tlie  fixed 
point  as  the  axes  of  x  and  y  respectively,  the  required  equation  is 
x2  +  (y  -  a)2  =  4?/2. 

Exercise  11.     Page  40. 

1.  X  —  ?/  +  1  =:  0. 

2.  2x  —  ?/  —  3  =  0. 

3.  X  +  ?/  -  1  =  0. 

4.  X  —  y  —  0, 

5.  3x  +  2?/  -  12  =  0. 

6.  2x  —  3?/  +  6  =  0. 

7.  X  +  1/  —  7  =  0. 

8.  4x  -  3?/  =  0. 

9.  y  =  o. 

10.  ?y==4. 

11.  6x-2y  =  0. 

12.  nx  —  m?/  =  0. 

13.  X  -  2/  -  3  =  0. 

14.  V3x-2/  +  7-2V3  =  0. 

15.  X  -  ?/  +  14  =  0. 

16.  V3x  +  3?/  +  12  -  13  V3  = 

17.  V3x-3i/-3V3  =  0. 

18.  X  +  2/  —  3  =  0. 

19.  V3x  +  2/  =  0. 


20. 

y  +  S  =  0. 

21. 

X  —  2  =  0. 

22. 

X  -  2/  +  2  =  0. 

23. 

X  —  y  +  ,5  =  0. 

24. 

X  —  y  —  4:  —  0. 

25. 

X  —  V.S?/  —  4  V3  =  0. 

26. 

y  +  4=0, 

27. 

V3x  —  2/  -  4  =  0. 

28. 

x  =  0. 

29. 

VSx  +  2/  +  4  =  0. 

30. 

X  +  2/  +  4  =  0. 

31. 

x  + V32/  +  4V3  =  0. 

32. 

2/  +  4  =  0. 

33. 

3x  +  iy  -  12  =  0. 

34. 

X  -  32/  +  6  =  0. 

35. 

X  +  2/  +  3  =  0. 

36. 

.3x  -  52/  -  15  =  0. 

37. 

X  —  22/  +  10  =  0. 

38. 

X  -  2/  -  1  =  0. 

b  ANALYTIC    GEOMETRY. 

39.  x  —  y  —  n  —  0. 

40.  4x  4-  2/  —  4n  =  0. 

41.  x  +  2/  -5V2  =0. 

42.  X-2/V3  +  10  =  0. 

43.  x  +  yVSH-  10  =  0. 

44.  X-2/V3-  10  =  0. 
x  +  7?/+  11  =0,  x-3y 
+  1  =  0,  3x  +  y- 7  =  0.  59.    (5,  -3),  (6,  4),  (-4,  -1). 


45, 
46 


52. 

y  -  -  X  ±  6  V2. 

53. 

•'     +  ^  -  1. 
11       11 

3        2 

55. 

C               6 

a  =  —  I  or 1 

A             m 

b 

_C 

b' 

58. 

A 
m  =  --,or- 

b^ 
a 

-i 

,  a;  -  Ty  =  39,  9x  -  5y  =  3,       60.    9x  +  2y  =  0,  |  V85. 
I      4x  +  ?/ =  11.  61.    ?/±x  =  yiqiXi. 


i  17x  —  3?/  =  25,  7x  +  9y  (  (d—c)x—{b—a)y=ad—bc, 

47.  ^  ^  62.    ^  ^  ^        ' 

(     =— 17,  5x-6?/  — 21  =  0.  I  (d-c)x  +  (6-a)?/=M— ac. 

r5x  — ?/  =  0,  5x  +  6?/— 35=0,  f2?/2a;+(xi— 2x2)2/— Xi7/2=0, 

48.  -^  3x  — 2/  =  21,  9x+4y  =  0,  64.   <i  2/2X+(2xi— X2)y— X]?/2=0, 

[2/  =  0,  14x  +  32/  =  29.  U2.c-(j;i+-C2)y=0. 

49.  X  -  2/  VS  —  7i  =  0.  65.    m  =  4. 

50.  2/  =  X  +  3.  66.    ??i  =  3. 

51.  2/ =  a;  ±  6  V2.  67.    6  =  -  9. 

68.   f^^  =  f^^ ,  or  xi  (2/2  -  2/3)  +  a;2  (2/3  -  2/i)  +  2:3  (2/1  -  2/2)  =  0. 

X3         Xi         X2         Xi 


Exercise  12.     Page  44. 

1.  -^3_Vl3x  +  T-\Vl32/  =  iiVl3;  p  =  iiVi3. 

2.  3\  Vsix  +  /^  V34y  =  If  V34  ;  p  =  if  V3i. 

3.  p  =  j2^Vl7.  9.   Fourth  quadrant. 

4.  p  =  ^"j  Vl3.  10.    Second  quadrant. 

5.  p  =  I V26.  11.    Fourth  quadrant. 

6.  p  =  \^  Vi.  12.    Second  quadrant. 
I 13.    Third  quadrant. 


7.  p  = 

8.  p  = 


Ve^  +  c2  14.    First  quadrant. 

r  15.    Second  quadrant. 

Vji-  +  c2  16.    Fourth  quadrant. 


ANSWERS.  7 

17.  Third  quadrant.  20.    m  = 

a 

18.  Fourth  quadrant.  21.    0  ;  8. 

22.    C=  12,  ^  =  4,  B=  -1. 

24.  ^  =  (2/2  —  2/i),  -S  =  —  (Xa  —  Xi),  G-  {xxVi  —  XiVx) 

25.  ,„^J^lI^^5:^^^yi-^i^''. 

X2        Xi  X2        Xi 

Exercise  13.     Page  46. 

1.  3x  —  2/  -  16  =  0.  5.    X  —  5  =  0. 

2.  3a;  -  42/ -  .3  =  0.  6.   x  +  4?/ +  49  =  0. 

3.  4x  —  ?/  =  0.  7.    7x  —  23?/  +  193  =  0. 

4.  2/  —  8  =  0.  8.    ?/  =  2x. 

9.  35?/  +  49x  -  79  =  0. 

Exercise  14.     Page  47. 

■n 

2.    tan  (i>  —  —  \.  3.    tan  <^  =  Jg.  4.    tan  <^ 


n2  +  2 

5.    90°.  6.    135°.  7.    90°.  8.    0°.  9.    30°. 

11.    j2/  =  5x— 10,  23_     (  ?/  — 3  =  m'(x  — 2),  _ 

Ix  +  5j/=28.  '     (  and7n'=  — (8±5V3). 

12     j2/=5x+ll,  [-2/  — 3=  ?n'(x— 1),_ 

'     (  X  +  52/  -  3  =  0.  14.   ^  ^      ,     8  ±  5  ^/3 

and  m— 

22.  2x  +  .3//  —  31  =  0.  ^  11 

23.  62x  + 312/ -1115  =  0.  r  x  -  .3?/ +  26  =  0, 

24.  2/-6X-27.  3°-   ^5^  +  3^+    8  =  0, 

, l2x  +  32/-    9  =  0. 

25.  2/  =  mx  ±  dVl  +  ?/i2. 

31.  X- 6  =  0. 

26.  £x  =  ^(2/-6).  r  2x-    92/4-    12  =  0, 

27.  ox  —  62/  =  a2  -  62.                              32.  J  lOx  —    42/  +    63  =  0, 

28.  (a±6)2/+ (6q:a)(x-a)=0.  Il8x  -  402/ +  111  =  0. 

rx—    ?/—    6  =  0^  meeting  in  the  point. 
33.  -J  2x  -    ?/  -    2  =  0  I         (-  4,  -  10). 
ISx  —  32/  -  10  =  0  J  Distance  =  V85. 

— -4±J5tan<^, 
3^-    '^-^^^=     i^±^tan0    (^-^^)- 


ANALYTIC    GEOMETRY. 


Exercise  15.     Page  52. 

3.    4.  4.    §V5.  5.    0. 

The  learner  should  construct  the 
given  Hnes,  and  observe  how  the  sign  of 
the  required  distance  gives  tlie  direc- 
tion of  tlie  point  from  the  line. 

8.   —0,  —5,  —  4,  3,  2,  1,  0,  —  1.     The  learner  should   construct 
the  lines,  and  observe  the  change  of  sign  of  the  distance,  as  in  No.  7. 


1. 

iVio. 

2.    fVs. 

7. 

-¥,-¥, 

16      12 

'5"»           "5'> 

-!-,  -h 

0,  +h 

-  hS  -  hS 

9         6 

5>             5> 

-1,      0, 

+  1- 

2  ^/a^  +  62 


Exercise  16.     Page  54. 

1.  H.  4.    40.  8.    35.  11.    26. 

2.  12.  5.    ah.  9.    lOf  12.    96. 

3.  29.  7.    26.  10.    i(xi?/2  -  a;22/i)-  13.    41. 

14.  i(a-c)(&-l).  21.   9a2.  27.    ^ 

15.  |(a-6)(a  +  6-2c).  g^.   f  ^g.   ,,, 

16.  i(a2-62). 

17.  60°,  00°,  00°;  9V3.  23.    24.  ^g     _^ 

18.  10.  24.    36. 

19.  j^.  25.    10.  30.    56. 

20.  H.  26.    iab.  31.    10^ 


2AB 


ANSWERS. 


Exercise    17.     Page  56. 
3.    2,  00,  90°,  2,  0°.  4.    0,  0,  45°,  0,  135°. 

5.    iV3-2,  2V3-I,  G0°,  — 1^^    150°. 

6.  2,  f  V3,  150°,  1,  00°.  26.    4y  =  x  +  8. 

7.  2,  -J^/3,  30°,  1,  300°.  27.    4?/  =  Ox -24. 

r-  OR      i  9x  —  20?/  +  0(3  =  0, 

8.  f  Vs.  -2,  00°,  1,  330°.  28.  /   .  .3,      ^ 
*       '         '        '    '                                 /  5x  —    4y  -f  32  =  0. 

9.  ill'J;+    2/  =  0,  29.    88x- 121;/ +  371  =  0. 
I      X  -  5//  +  20  =  0.  g^^  5^  _    y  _  10  =  0, 

10.  /i  V827  '    ^  +  '">?/  —  -8  =  0. 

3x  +  4//  -  57  =  0,  31.    j  -•«  +    2/  -    *^  =  <^' 

3x  +  4(/+    0  =  0,  '    X- 2// -17  =  0. 

11.  ■{  12x  —  5//  —  30  =  0,  32  \  4x  +  ?/  -  20  =  0, 
12x  -  5y  +  24  =  0.  '  I  X  —  4?/  -  5  =  0. 
Area  =  63. 

12.  43. 

13.  x=3. 

jx-y+l=0, 
U+?/-7  =  0. 

15.  5x  +  G?/  —  39  =  0. 

16.  14x  -  3?/ -  30  =  0. 

17.  4x  —  5y  +  8  =  0. 

18.  X  +  //  —  7  =  0. 

■     X  —  X3       X2  —  Xi 

20.  i  2^  =  •^'  1-'^  =  5x  -  1, 
19?/=  5x  +  7. 

21.  92x  +09//+  102  =  0. 

22.  x  +  4?/  =  34. 

23.  3x  +  4?/  —  5(1  =  0. 

24.  3x  +  4?/  =  24. 

25.  2/ -2/1=  -^(x-Xi). 

Xi 


33. 

2x  =  y,  2?/  =  X. 

34. 

4x  +  5//+  11  ±3  V4r  =  0. 

35. 

2/  =  (7  q:  5  -V^)  (X  +  2). 

36. 

2x  —  5?/  _   I  4x  +  3//  —  12 

V29                         5 

37. 

^  7X-32/+  15  =  0, 
1  3x  +  7//  —  03  =  0. 

38. 

\    8x  +  7//  -19  =  0, 
I  16x  +  3y  +17=0. 

39. 

135°. 

40. 

00°. 

41. 

31  V20 
143 

42. 

.  hh  +  al(^  —  ah 
\/a^  +  62 

43. 

c- 

V/i^  +  t-i 

44. 

±^Vl  +  m--'. 

45. 

C-. 

10  ANALYTIC    GEOMETRY. 

xy  represents  the  two  axes. 

a  =  5. 

x  +  a  =  0,  X  —  b  =  0. 

x  +  a  =  0,y+  b  =  0.' 

The  axes  and  x  =  y. 

2x  —  y  =  0,  7x  +  y  =  0. 

62.  If  h  denotes  the  altitude  of  the  triangle,  and  the  base  is  taken 
as  the  axis  of  x,  the  locus  is  the  straight  line  y  =  h. 

63.  The  equation  of  the  locus  is 

(x  —  Xi)2  +  {y-  yi)-  -  {X—  iCo)^  +  {y  —  y^f. 
Tills  is  the  equation  of  the  straight  line  bisecting  the  line  joining 
(Xi,  2/i)  and  (X2,  y^)-,  and  _L  to  it. 

64.  The  two  parallel  lines  represented  by 


46. 

fe2 

6' 

2a2  + 

54. 

47. 

•  bab  +  262 
6 

57. 
58. 

48. 
49. 

17i. 

59. 

50. 

59. 

60. 

51. 

(10,5 

\)- 

61. 

Ax+By-^C±d  ^A^  +  i'-'  =  0. 

__         ,      _,                   __     Ax  +  By+C  ,   A'x+  B'y  +  C      , 
65.   x  +  y  =  k.  66.    ,  — I ^-  =  k. 

Vyl2  +  B^  sJA'-^  +  B'2 

67.  Let  h  denote  the  base,  k"^  the  constant  difference  of  the  squares  of 
the  other  two  sides.  Taking  the  base  as  the  axis  of  x,  and  the  middle 
point  of  the  base  as  origin,  the  equation  of  the  locus  is  26x  =  ±  fc^. 

Exercise  18.     Page  64. 


1. 

lx  +  y-  0. 

4 

J  X  -  ?/  +  8  =  0, 

6. 

64x  -  23?/  =  59. 

2. 

X  +  2?/  -  13  =  0. 

\  x  +  ?/_6  =  0. 

7. 

44x  +  ?/  =  0. 

3. 

bx  +  Qy  —  37  =  0. 

5. 

2/  =  X  +  3. 

8. 

5x  +  2/  -  16  =  0. 

9.    {A  C  -  A'C)x  +  (BC  -  B'C)ii  -  0. 
10.    {BA'  -  AB')y  +  CA'  -  AC  =  0. 
Ax+By+C  _   A'x  +  B'y  +  C 
■    Axi  +  Byi+  C~  A'xi  +B'yi  +  C' 
12.    472x  -  292,'  +  174  =  0.  13.    2/  =  x  Vs  +  3  -  V3. 

(4x+32/—  25  =  0,  ^_^  —  "^^~  ^ 

'     (  3x  —  42/  +  25  =  0.  'a        &      ma  +  6 

16-18.  Generally  the  easiest  way  to  solve  such  exercises  as  these 
is  to  find  the  intersection  of  two  of  the  lines,  and  then  substitute  its 
coordinates  in  the  equation  of  the  third  line. 


ANSWERS.  11 

-^             -                                   ^«     ,TTi        '>^"  ~  '>n       b"  —  b 
19.    m  =  1.  20.    When  — =  — • 

in    —  m       b    —  b 

21.  If  we  choose  as  axes  one  side  of  the  triangle  and  the  corre- 
sponding altitude,  we  may  represent  the  three  vertices  by  (a,  0), 
(-  c,  0),  (0,  b). 

22.  Choosing  as  axes  one  side  and  the  perpendicular  erected  at  its 
middle  point,  the  vertices  may  be  represented  by  (a,  0),  (—  a,  0),  (6,  c). 

23.  It  is  well  here  to  choose  the  same  axes  as  in  No.  21. 

24.  Choosing  the  origin  anywhere  within  the  triangle,  it  is  evident 
that  the  equations  of  the  bisectors  in  the  normal  form  may  be  written 
as  follows : 

(X  cos  a  +  y  sin  a.  —  p)  —  {x  COS  a   +  y  sin  a   —  p')  =  0, 

(x  cos  a  +  y  sin  a.'  —  p')  —  (x  cos  a."  +  y  sin  a"  —  p")  —  0, 
(x  cos  a"  +  y  sin  a"  —  p")  —  (x  cos  a  +  y  sin  a.—  p)      =  0. 
Now,  by  adding  any  two  of  these  equations,  we  obtain  the  third  ; 
therefore,  the  three  bisectors  must  pass  through  one  point.     • 

2g_    (  2V2,  VlO,  2V1O.  29     (x-2/  +  2  =  0 


Origin  within  the  A-  (x  +  y  —  14  =  0. 

26.    iJ- VlO,  1  & V.34,  If  V]3. 

(X  —  1=0 

27  (^  +  2/  +  10  =  0,  30.    ^_i^o: 

I  7x  -  7?/ +  24  =  0. 

28  i ''*  ~  9?/ +  34  =  0,         g^    ±(y— ?»x— b)  _^  ±(y—jn'x-b') 
'    \  9x  +  ly  -  12  =  0.  ■      Vl  +  ?n2  Vl  +  m'2 


Exercise  19.     Page  68. 

1.  (i)  Parallel  to  the  axis  of  x,   (ii)  parallel  to  the  axis  of  y. 

2.  When  ad  =  be. 

3.  The  two  lines  are  real,  imaginary,  or  coincident,  according  as 
C'^  —  iAB  is  positive,  negative,  or  zero.  The  two  lines  are  _L  to  each 
other  when  A  +  B  =  0. 

5.  X  +  y  +  1  =  0,  and  x  —  3?/  +  1  =  0. 

6.  x-2y  ±(y  -  3)  V^  =  0. 

7.  X  —  y  -  3  =  0,  and  x  —  3y  +  3  =  0.         8.  45°.         9.  K  =  2. 
10.  K  =  -  10,  or  —  ?./  .  11.    K  =  28.  12.    A'  =  V-. 


12  ANALYTIC    GEOMETRY. 


Exercise  20.     Page  70. 

1.  Take  the  point  O  as  origin,  and  the  a.xiss  of  ij  parallel  to  the  given 
lines.  If  the  ecjiiations  of  the  given  lines  are  x  =  a,  x  —  b,  and  if  the 
slopes  of  the  lines  drawn  in  the  two  fixed  directions  are  denoted  by 
m',  jn",  the  eiiuation  of  the  locus  is 

{b  —  a)  1/  =  vi'b  (x  —  a)  —  m"a  (x  —  b). 

2.  Jf  a  and  b  are  the  sides  of  the  right  triangle,  the  equation  of  the 
locus  is  f( 

3.  Let  OA  =  a,  OB  =  b.     Then  the  equation  of  the  locus  is 

X  +  y  —  a  +  b. 

4.  Take  as  axes  the  base  and  the  altitude  of  the  triangle.  Let  a 
and  b  denote  the  segments  of  the  base,  h  the  altitude.  Then  the  equa- 
tion of  the  locus  is  2x         2?/ 

b  —  a        h 
This  is  a  straight  line  joining  the  middle  points  of  the  base  and  the 
altitude. 

5.  Take  as  axes  the  sides  of  the  rectangle,  and  let  a,  b  denote  their 
lengths.     The  equation  of  the  locus  is 

bx  —  ay  =  0. 
Hence,  the  locus  is  a  diagonal  of  the  rectangle. 

Exercise  21.  Page  73. 

1.  X-  +  y-=  —  2rx.  13.  (4,  0),  4. 

2.  x-2  +  2/2  =  2ry.  14.  (-  4,  0),  4. 

3.  x-^  +  y'^=  -  '-iry.  15.  (0,  4),  4. 

4.  (X  -  5)-  -¥  {y  +  ;])-  =  100.  16.  (0,  -  4),  4. 

5.  X--'  +  (//  +  2)-^  =  121.  17.  (0,  I),  1. 

6.  (X  -  r.)-;  +  y-'  =  2.5.  18.  (0,  0),  3A;. 

7.  (X  +  .5)-'  +  y-  =  25.  19.  (0,  0),  2k. 

8.  (x  -  2)2  -\-{y  -  Zy^  =  25.  20.  (0,  0),  VoM^. 

9.  x2  +  2/2  _  .2hx  -  2ky  =  0.  ^l.  (-,  o\  -  VH. 

11.    (1,2),V5.  ^'^       ^    ^. 

^       '  'h   k\     VA2  +  k^ 


12.    a,  I),  ■V(J2.  22.    (^,-), 


2 


ANSWERS.  13 

23.  When  D  =  D'  and  E  =  E' ;  in  otliiT  words,  when  the  two 
equations  differ  only  in  their  constant  terms. 

24.  In  this  case,  r  =  0.  Hence,  the  equation  represents  simply  the 
point  (a,  b).  We  may  also  say  that  it  is  the  equation  of  an  inlinilely 
small  circle,  having  this  point  for  centre. 

{{%,  I),  |V2;  r   (i)  Z>2-4c. 

26.  <;  On  OX,  3  and  2 ;  31.  \   (ii)  E^  =  4C. 

lOn  OY,  6  and  1.  y  (iii)  4C >  D'^  and  E^. 

( (6,  2),  5 ;  32.    x-2  +  2/2  +  iqx  +  10?/  +  25  =  0. 

27.  ^  On  OX,  6±V21; 

[On  or,  imaginary  point.s.  ^^-    (''  "*)  =^'"M8,  1). 
r(2,  4),  2V5;  34.    (2,  0)and(^,  -  ^). 

28.  -^'  On  OA',  0  and  4 ;  35.    i  V5. 

I  On  OY,  0  and  8. 


r(3,  -2),3; 


36. 

V         a^  +  bO 

37. 

•2x  -  2/  -  2  =  0. 

38. 

4.C  — 5?/  — 71  =  0. 

39. 

3x  -  5?/  -  34  =  0. 

29.  -^  On  OX,  3  ±  V5  ; 

[On  OF,  -2. 
r(-ll,  9),  VhB; 

30.  -^  On  OX,  -  3  and  -  10 ; 

I  On  OY,  9±2V6. 

40.  Let  {x,  y)  be  any  point  in  the  required  locus ;  then  the  distance 
of  {x,  y)  from  (Xi,  2/1)  must  always  be  equal  to  its  distance  from  (jo,  2/2)  ; 
therefore,         (x  —  Xi)2  +  {y  —  2/1)2  =  (x  —  X2)2  +  {y  —  7/0)- ; 

whence  2x  (xi  —  Xo)  +  2y  (2/1  —  2/2)  =  (-Ci"^  +  yr  —x-z^  —  2/2'-). 

Show  that  this  represents  a  straight  line  _L  to  the  line  joining  (xi,  2/1) 
and  (xo,  2/2)  at  its  middle  point. 

41.  8x  +  62/  +  17  =  0. 

42.  First  Method.  Substitute  successively  the  coordinates  of  the 
given  points  in  the  general  equation  oi  the  circle ;  this  gives  three  equa- 
tions of  condition,  and  by  solving  them  we  find  the  values  of  a,  b,  r. 

Second  Methoi>.  Join  (4,  0)  to  (0,  4)  and  also  to  (0,  4)  by  straight 
lines,  then  erect  perpendiculars  at  the  middle  points  of  these  two  lines ; 
their  intersection  will  be  the  centre  of  the  circle,  and  the  distance  from 
the  centre  to  either  one  of  the  given  points  will  be  the  radius. 

A ns.  K'-  +  y'  —  fix  —  %  +  8  =  0. 

43.  x2  +  y^  —  8x  +  62/  =  0.  45.    x-  +  y-  +  Sax  -  6ay  =  0. 

44.  x2  +  2/2  +  Ox  +  2/  =  0.  46.    x2  +  2/2  +  8x  +  20 y  +  31  =  0. 


14  ANALYTIC    GEOMETRY. 


48. 
49. 


47.  x'-  + ?/-  — Ox  — %+ 14  =  0.  51.  x-  +  y-::f2ax^:2ay  +  a'^=0. 

^  (X  -    o)--^  +  (>j  +  8)--2  =  169,  52.  x2  +  y^  =ax  +  by. 

]  (X  -  22)-^  +  {y-  9)-  =  169.  53.  (x  -  1)2  +  {y  —  4)^  =  20. 

<  x-2  +2/2  _oo(a;  +y)  +225  =0,  54-  «-  +  2/'  -  14x  -  4?/  -  5  =  0. 

Ix'+y-—  (}(x+y)+     9=0.  55.  x'- +  ?/2  ±  V2a?/ =  0, 

50.  x2  +  2/-  —  8x  —  8y  +  16  =  0.  X--  +  y-±  \^ax  =  0. 
56.    r?((x'+?/2)— a6=  (ma— 6)x+  (m6  —a)?/.    57.   x-  +  ?/2  =  XiX  +  yiy. 

58.  (X-Xi)(X-X3)+(2/-yi)(2/-2/2)=0.  3.2_„j,+y2  =  r2_  ^" 

59.  (l  +  m2)(x2+2/^) -2r(x+m2/)  =  0.  *  ^  2" 


Exercise  22.     Page  81. 

1.  The  double  sign  corresponds  to  the  geometric  fact  that  two  tangents 

having  the  same  direction  may  always  be  drawn  to  a  given  circle. 

3.  2x  +  3y  =  26,  3x  —  2y  =  0;  sVlS,  2VT3,  -9,-4,  -L3 Vl^ 

Xi2  — r2  r3  „„     rx2  +  ?/2  =  p2^ 

'     —  ^'     •  22.  .;  ,  .      , 

Xi2/i  (  (p  cos  a,  p  sill  a). 

r  When  C  =  rV^lM^. 

23.  J  rxTu      Aa+Bb-C       , 
]  When  — ,  —  =  ±r. 

[  V^2  +  J52 

24.  ax  +  by  =  0. 

25.  (—  a,  —  b). 

26.  (2a,  b). 

27.  (0,  6). 

28.  x2  +  y2  =  |. 

29.  m  =  0. 

30.  c  =  —  36  +  20V6. 

14.  Ax  +  By  If  W^2  +  £2  :=  0.     31.    (x  -  5)2  +  {y  -  3)2  =  Jj-y-- 

15.  5x-^2/=fWI^T^  =  0.     32     ((X-    2)2+ (.y-    4)2=100, 

,  fi  ..  -  .  +  .V2  =  0.  i  (X  - 18)^  +{y-  li>)'  =  100. 


Xi 

5. 

9x  -  13y  =  250. 

6. 

x±Sy=  10. 

7. 

104 1. 

8. 

x2  +  2/2  =  25*. 

9. 

14x  ±6y  =  232. 

10. 

3x  +  2/  =  19. 

11. 

3x  +  42/  =  0. 

L2. 

(  3x  +  72/  =  93, 
(  3x  -  7y  =  65. 

13. 

X  =  r. 

16.  x-2/±r\/2  =  0 

17.  The  equation  of  the  two  taii- 


33.   (X-  1)2+  (2/ -6)2  =  25. 


gents  is  {h'-  -  r^)y2  =  r2(x  -  /02.  34.  -=-+-■ 

18.  x  +  y=  ±r\f2.  35  (3.2  -\-y2^a  +  b  +  Va2  +  62)2. 

19     (a;  =10,  -2a6(a  +  6  + Va2  +  &2)(x+y) 

(  3x  +  42/  =  50.     _  -f  ^252  =  0. 

20.  2/  =  2x  +  13  ±  6  V5.  36.  x  =  a  +  r. 

21.  -21,  -3f.  37.  [4r2-2(a-6)2]i. 


ANSWERS.  15 


Exercise  23.     Page  84. 

1.  iV29,   (1,  -  1).  11-  ^'  +  2/-  -  5x  -  I2y  =  0. 

2.  |Vrr,  (-  h  f).  12.  x2  +  2/2  -  14x  -  4?/  -  5  =  0. 

3.  iVsi,  (I,  ^).  13.  x2  +  ?/2  +  i4j.  +  ]4y  +  49  =  q. 

/        6  ab      \  14.  x2  +  ?/■- ;^  2rx  —  2ry  +  r2  =  0, 

*■    ^'VVTTa^'     VrTa2/  x2  +  r' ±  2rx  +  2ri/ +  r2  =  0. 

5.  x"-  +  y^-Sl.  15.  x2  +  2/2-2ax-2a2/+a-'--=0. 

6.  (X-  7)2 +  2/2  =9.  ;Lg_    3.2  +  ^2=:  9. 

7.  (X  +  2)2  +  (y-  5)2  =  100. 

o       9.0      o   /o     /^   \-rt         17.   5(x2+2/2)-10x,+  30?/+49=0. 

8.  x2  +  2/2  —  2a(3x  +  4?/)  =:  0.  ^        ^  '  •' 

9.  x2  +  2/2  +  262  4-  c-2  18     H^  ~    ^^^  +  ^^  ~    ^)^  ~  ^' 
=  2  [(6  +  c)  X  +  (6  -  c)2/] .  ■     (  (X  +  V)'  +  (y  +  V)'  =  f  • 

10.    3a6(x2  +  2/2)  +  2a6(a2  +  62)       19.  x2  +  y"-  —  30x  —  Uly  =  0. 

=  (5a2+262)6x+  {ob-^+2a-)ai/.    20.  x2  +  2/2  +  SOx  +  882/  —  50  =  0. 

2^      {      x2+      2/2  — .3Gx-    402/ +324  =  0, 
I  25x2  +  25^2  _  80x  -  4942/  +    64  =  0. 

22.  (6,2),  5.  31.  x2+2/2  =  ±a2/V2,or  =  ±axV^ 

23.  -  15.  32.  x2  +  2/2  ±  2a(x  ±  2/)  =  0. 

24.  —  10.  33.  2(x2  —  ax  +  2/2  -  r-)  +  a2  =  0. 

25.  1V26.  34.  X  — y  =  0. 

26.  VTO.  35.  4x  +  32/=^0. 

27.  x,x  +  y,y  =  x,^  +  2/,2.  36.  (18  ±  2V41)x  -52/  =  0. 

28.  (i)  Z»2  =  A  AC,  (il)  -B2  =  4^(7,  ^'^^  a:  +  "^^  ±  ^0  -  0. 

(iii)  Z>2  =::  £2  rr  4^  C.  38.    X  +  ^  -  10  =  0. 

29.  r2  =  2rmc  +  c2.  40.  +(35  +  24\/30).     43.   135°. 

30.  k  =  40,  or  -  10.  44.   (7,  -  5)  and  (-  6^^,  9||). 

f  (X  +  4)2  +  (2/  +  10)2  =  85, 
«.    ^/^_514X2      /         670y^_8^ 
[\         109/       V^      169/       1692 

46.  The  circle  (x  —  Xi)2  +  (2/  —  Vi)-  ~  r". 

47.  The  circle  (x  -  a)-  +  (2/  —  6)2   =  (r  +Y)'^, 

or  (X  -  a)2  +  (2/  -  6)2  ^  (r  -  r')2. 

48.  The  circle  (x  —  rt)2  +  (2/  —  6)2   =  7-2  +  <2. 


16  ANALYTIC    GKOMKTKY. 

49.  Take  yl  as  orijiin,  and  let  the  radius  of  the  circle  =  r;  then  the 
locus  is  the  circle  x-  +  //'-  =  rz. 

50.  Take  A  as  origin,  and  let  the  radius  of  the  circle  =  r  ;  then  the 

.    ,      „  ,      -,       2»irx 
locus  IS  the  circle  x-  +  y'^  =  - — , — 
m  +  n 

51.  Take  A  as  origin,  AB  as  axis  of  x,  and  let  AB  =  a  ;  then  the 
locus  is  the  circle  {in-  —  n-)  (x'^  +  y'^}  —  2am^  +  a-m-  =  0. 

52.  Take  AB  as  the  axis  of  x,  the  middle  point  oi  AB  as  origin, 
and  let  AB  =  2a  ;  then  the  locus  is  the  circle  2{x^  +  y-)  —  k- —  2(jfi. 

53.  Using  the  same  notation  as  m  No.  52,  the  locus  is  the  straight 
line  4ax  =  ±  Ic^. 

54.  Taking  the  fixed  lines  as  axes,  the  locus  isthecircle4(x-+?/-)  =  fZ-. 

55.  Take  the  base  as  axis  of  x,  its  middle  point  as  oi-igin,  and  let 
the  length  of  the  base  =  2a,  and  the  constant  angle  at  the  vertex  =  d. 
Then  the  locus  is  the  circle  x'^  +  y-  —  2a  cot  dy  =  a'^. 

56.  Take  A  as  origin,  AB  an  axis  of  x,  and  let  AB  —  a,  AC  =  h. 

Then  the  locns  is  the  circle  (x  —  },a)'-  +  y-  =  —• 

-   '        ■         4 

4r* 

57.  The  circle  x-  +  y-  =  r-; zzi  where  I  is  the  length  of  the  chord. 

58.  The  locns  is  a  circle. 


Exercise  24.     Page  97. 

1.  7x  —  6?/  =  0.  2.  X  —  ?/  =  0. 

4.  X  +  ?/  =  r,  2x  +  o//  =  r,  (n  +  b)x  +  (a  —  b)  y  =  r^. 

5.  l?.x  +  2>/  =  40.  6.  The  tangent  at  {h,  k). 

7.  (i)  2x+%  =  4,   (ii)  '^x  —  y  =  4,  (iii)  x  — ?/=4. 

8.  (i)   (20,  ;]n),  (ii)   (21,  -14),  (iii)  (.%«,  nr>h). 

9.  (G,  8).  18.   12x  +  17;/  -  51  =  0. 

^n  ( --^     -JJ^1\  19.  x+ 7/ -2  =  0. 

"•  V      c  '        c  ) 

11.  (4,  ±  3),  4x  ±  ?,y  =  25.  ^^    ("'  ~"^^^^  ~("^  -lfi)y  +  ac  =0. 

16.  h:2  +  /,-2  _  r2.  21.  X  -  7/  =  0,  Vi(a  +  6)^  -  4c. 

17.  3.  22.   (-2,  -1). 


ANSWERS.  17 

Exercise  26.     Page  109. 

1.  Writing  x  +  1  for  x,  and  y  —  'l  for  ?/,  and  reducing,  we  have  y'^—^x. 

2.  x-  +  (/-  =  ?•-.  5.   x'^  +  ?/"  =  '''■'■ 

3.  x'-^  +  y-  =  2rx.  6.   2xy  =  a^. 

4.  X-  +  y-  =  —  2ry.  7.   x-  —  //-  +  2  =  0. 

8.  (i)  p  =  ±  ((,  (ii)  p2  cos  2d  -  a^. 

9.  (i)  p  =  ia  tan  0  .sec  ^,  (ii)  (a  +  p  co.s  ^)-  =  4ap  sin  ^. 

10.  (i)  X-'  +  //■-  =  a-,  (ii)  x-  +  y-  =  rtx,  (iii)  x^  —  y-  —  ofi. 

11.  x  +  y  =  ().  15.  x2  —  ijxy  -\-y-  =  0. 

12.  2x  —  5y  +  10  =:  0.  16.   xy  =  .'J. 

13.  12x-  +  Ulzy  +  4i/^  +1  =  0.  17.    y/-  =  2(f(x  V2  —  a). 

14.  X-  +  y-  —  25.  18.    -Ix//  =  25. 

Exercise  27.     Page  111. 


1. 

6V3. 

12. 

9x2  +  25(/2  =:  225. 

2. 

4  sin  ^w. 

13. 

14. 
15. 

p=  8a  cos  e. 

p  =  ±  4a. 

p2  sin2  e  —  5/3  cos  0  =  -2:^. 

3. 

Vl3  —  12  cos  w. 

4. 

Va-  +  Ij^  —  2ab  cos 

(^- 

0). 

5. 

2a  sin  0. 

16. 

p2  =  49  sec  20. 

6. 

2(1  cos  0. 

17. 
18. 

P'  =  /i;2  cos  20. 
xy  -  a-. 

7. 

aV5-2V3. 

9. 

2x'-2  +  2x?/  +  2/2  =  1. 

19. 

{x-  +  2/2)2  =:  2ixy. 

10. 

2x2  +  2/2  =  6. 

20. 

^s— 2/3+  (3x— Sy— 5A;)x2/= 

=0. 

11. 

2/  =  0. 

21. 

tan-i  |. 

22.    (i 

)  tan 

-(■ 

-iy 

(ii)  tan-'-- 
.4 

Exercise  28.     Page  117. 

2.   2/2  =  4/y(x  -  7/).  3.   2/- =  ^^(x  +  p). 

4.  (i)  ^2  =  lOx,  (ii)  y-  =  lOx  +  2-'.,  (iii)  2/2  =  lOx  —  25. 

5.  (i)  2/2  =l(3x,  (ii)  y-  =  lOx  +  04,  (iii)  y-  =  lOx  —  G4. 


18  ANALYTIC    GEOMETRY. 

6.  (0,  0),  (2,  6).  8.    (4,  6)  and  (25,  15). 

7.  6,  15,  ^-  9.    (12,  6). 

10.  The  line  x  =  9  meets  the  parabola  in  (9,  6)  and  (9,  —  6).  The 
line  X  =0  passes  through  the  vertex.  The  line  x  =  —  2  does  not  meet 
the  parabola. 

11.  The  line  y  —  6  meets  the  parabola  in  (9,  6).  The  line  y=  —8 
meets  the  parabola  in  (16,  —  8). 

12.  p  =  4.  13.    (0,  0),  (2,  8). 

14.  (i)  y  =  0,  (ii)  x  =  -  2,  (iii)  x  =  2,  (iv)  4x  ±  3y-8=0,  (v)  y=  -  2x. 

15.  (i)  4x  -  5?/  +  24  =  0,  (ii)  x^  +  y^  -  20x  =  0. 

16.  3j3.  17.   8pV3. 

24.  The  latus  rectum  of  each  =  4p.  The  common  vertex  is  at  the 
origin.  The  axis  of  x  is  the  axis  of  (i)  and  (ii);  that  of  y  is  the  axis  of 
(iii)  and  (iv).  Parabola  (i)  lies  wholly  to  the  right  of  the  origin, 
(ii)  wholly  to  the  left,  (iii)  wholly  above,  (iv)  wholly  beloiv.  We  may 
name  them  as  follows  : 

(i)  is  a  right-handed  X-parabola.      (iii)  is  an  upward  F-parabola. 
(ii)  is  a  left-handed  X-parabola.         (iv)  is  a  downward  F-parabola. 

Exercise  29.     Page  121. 

6.  X  —  4?/  -I-  20  =  0,  4x  +  y  —  90  =  0. 

_^  (x-?/  +  3  =  0,  ,    <x  +  y-9  =  0, 

7.  Tangents  -{      ,       ,   „      „      normals  ' 


•^1: 


x-{-y  +  ^  =  0;  lx-y-Q  =  0. 

These  lines  enclose  a  square  whose  area  =  72. 
8.   Tangent  =  V266,  normal^ V95,  subtangent  =  14,  subnormal^ 5. 

9-    (5,  10).  13.   S^^,    I^Vr^M^. 

in  Vl  +  m^      "i 

14.  [sjx.xo,  l{y,  +2/2)]. 

15.  x  +  y  +  p  =  a,  point  of  contact  (p,  —  2p),  intercept  —  —  p. 

16.  Equationsof  the  tangents  ?/V3  =  ±x±3p,  required  point(—3p,0). 

17.  For  the  two  points  whose  coordinates  are 

X = f  (1  ±  vi7),  2/ =  ±  p  Jl±^. 

o  \         2 

18.  For  the  points  (0,  0)  and  (3p,  ±  2j5  VS). 

19.  9x  -  62/  +  5  =  0,  (I,  I). 

20.  X  -  2?/  +  12  =  0,  (12,  12). 
4x  +  2y  -h3  =  0,  (f,  -  3). 


ANSWERS. 


19 


21.    ?/  =  x(±V2- 1)  +  4(±  V2  + 1). 


22.     4Vp(p  +  Xi)8 

Xi 

24.    By  the  secant  method  we  find  that  the  equation  of  the  tangent 
at  (xi,  ?/i)  is  y—  y\  _      4 

a:  —  a;i      2/i  —  3 

The  points  of  contact  are  (—1,11)  and  (  —  1,-5)  ;  hence  the  tangents 
are  z  —  'ly  -\-  23  =  0 

and  X  +  2i/  +  11  =  0. 

r    (i)  2/12/  =  -  2p(x  +  xi), 

25.   ^   (ii)  xix  =       2i)(?/  +  t/i), 

[(iii)  Xix=  —  2i3(?/ +  ?/i). 


Exercise  30.     Page  123. 


1. 

4. 

7. 

10. 


t/2  =  24x  —  144.  2.    2/2  =  i(5x. 

j  2?/2—  lix+  12?/ +  73  =  0, 
j  2y2+  llx+  12y-37  =  0. 
2x2  =  Qy.  8.    I,  8x  +  3 


3.    2/2=  -17x. 

5.  (2/ +  7)2  =  4(x  -  3). 

6.  32/2  =  4x. 

0,  8x  ±  \by  -3  =  0. 


4  on  OX ; 

8  and  —  2  on  OY. 

11.  4(2  ±  V3)p. 

r    (i)2/  =  x+2, 

12.  \    (ii)  -2^, 

[(ill)  x+  2/-G  =  0. 

13.  x  +  y-6  =  0. 

2p 

14.  y  -  2/1  =       (X  -  xi). 

2/1 

15.  (8,  4),  (2,  10). 

16.  (2,  4),  (11,  10). 
_  b  ±  \lh'  +  4ap  ^ 


20. 

y"=-  9x. 

21. 

2/2  =  8x. 

22. 

4r2  -  «2 

23. 

y2=  —X. 

24. 

2jl2 

y--    . 

17.    2/ 


2a 


Vn2  +  «2 

25.  2/2  =  2(2r- s)x. 

26.  4pV2. 

27.  The  equation  of  the  circle  is 

(X- 3)2+  (2,-1)2=2^5. 

28.  (-3p,  0). 


'  A  left-handed  A'-paiabola.   29.    f-,  ±?pV3Y 
Latus  rectum  =2.  ^  ' 

30.  j(^'±2p); 
I  45°  and  135°. 

31.  4p-. 
34.    The  parabola  y- 


18.  <|  Vertex, (—2,  0). 
Focus,  (—  5,  0). 
Directrix,  x  = 

19.  —  2aV2. 


px. 


20  ANALYTIC    GEOMETRY. 

The  loci  in  Nos.  .35—38  are  parabolas,  the  latus  rectum  in  each 
being  half  that  of  the  given  parabola.  If  the  given  parabola  is  y"^  =  4px, 
the  equations  of  the  loci  are  :  39,    Tj^g  straiglit  line  y  =  pk. 

35.  ?/2  =  2px  —  p2.  4Q     ^,^g  parabola  7/^  -  4px  =  p-k^. 

36.  ?/-2  =  2px  -  2p2.  4f,    The  straight  line  A:x  -  p. 

00'    K  ~  ?^' ,  o  0  42.    The  circle  (x  -  p)2  +  ^2  =  ^. 

38.    ?/2  =  2px  +  2p2.  ^        ^'    •  "       jc^ 

43.    Take  the  given  line  as  the  axis  of  ?/,  and  a  perpendicular 

through  the  given  point  as  the  axis  of.x,  and  let  the  distance  from  the 

point  to  the  line  =  a.     The  locus  is  the  parabola  y-  =  2a(x  —  -Y 

Exercise  31.     Page  134. 

10.  3x-5y-6  =  0.  fay  =  2px. 

11.  8y  —  25=  0.  14     J  T''^^  chord  is  parallel  to  the 

I      tangent  at  the  end  of  the 

12.  13x  +  22y  +  k  =  0.  ,.        , 

-'  1^     diameter. 

13.  X  —  7/  —  1  =  0.  15.    ?/-  =  52x. 

18.  Writing  the  equation  in  the  form  (y  —  3)-  =  8(x  —  2),  and  pass- 
ing to  parallel  axes  through  (2,  3),  we  have  y^  =  8x.  8,  (2,  3),  (4,  3), 
V  =  3,  X  =  0. 


19.    A  numerically,  (  — —, — » ^  r  2/  =  —  „ 

^'  \      4A  2)  2 


B2-4C  B\  B 

4J~'    --2>^=-2- 

r.«     „             ■     n      /       ^    A^-4C\  A 

20.    />  numencallv,  I  — „»  r^:: —  I'  x=  —  — • 


.     ..       /      A    A-^-^V\ 


21.  Take  the  given  line  as  the  axis  of  y,  and  a  perpendicular  through 
the  centre  of  the  given  circle  as  the  axis  of  x.  Let  the  radius  of  the 
circle  =  r;  distance  from  the  centre  to  the  given  line  =  a.  There  are 
two  cases  to  consider,  since  the  circles  may  touch  the  given  circle  either 
externally  or  internally.     The  two  loci  are  the  parabolas 

?/2  =  2{a  ±  r)  X  +  r-  —  o-. 

22.  Let  2a  be  the  given  base,  ah  the  given  area  ;  take  the  base  as 
axis  of  X,  its  middle  point  as  origin  ;  then  the  locus  is  the  parabola 

x2  +  by  =  a?. 

Exercise  32.     Page  144. 
1.    5,  4,  3,  |.  2.     V2,  1,  1,  iV2.  3.    2,  ^/3,  1,  \. 


1_^    J^^        Is- A         IB-A 

■\Ia  '    Vi>'' 


"^B  V^'  V^'^^^'"^^^- 


SWERS. 

13. 

x'-i 

+  2(/2  = 

;  100. 

14. 

8x2  +  y?/2  = 

=  8tt2. 

16. 

2 : 

V3. 

17 

X 

=  y=± 

«6 

■  V«' + ^- 

18. 

(1,  t)^  (- 

S      —    2\ 

31         sr 

19. 

(1 

,  -0,  (1, 

-2). 

21 

5.  fVe. 

6.  e  =  iV3. 

7.  4x2  +  9y2  =  144, 

8.  25x2  +  1097/2  =  4225. 

9.  16x2  +  252/2  =  .3600. 

10.  16x2  4-252/2=  1600. 

11.  25x2  +  1692/2  =  4225. 

12.  3x2  +  72/2=115.  20.    (:!,  l),(3,-l),(-3,  l),(-3,- 1). 

21.  The  equation  of  the  locus  is  x2  +  42/2  =  r-. 

22.  Taking  as  axes  the  two  fixed  lines,  and  putting  AP  —  a,  BP  =  6, 
the  acute  angle  between  AB  and  the  axis  of  x  =  ^,  we  find  that 

X  —  a  cos  <p,  y  =  b  sin  (p. 
Therefore  P  describes  an  ellipse  whose  equation  is 

^  +  ^  =  1. 
a2      62 

23.  The  two  straight  lines  y  =  ±  •'^\l~  r-    "^^^  locus  is  imaginary 
when  y  is  imaginary;   that  is,  when  A  and  B  have  like  signs. 

29.   The  equations  of  the  sides  are 

,       ab  ,       ab 

X  =  ±— 1   2/=± 


M 


Va2+62  ^a;^+l^' 

4a262 
area  =  ,-,  ,      • 

Exercise  33.     Page  151. 

4x±9y  =  35, 


'■■\ 


7. 

22/  =  X  ±  10. 

8. 

4x  -  32/  ±  Vl07  =  0. 

9. 

X  =  ±     .  ^  _  '  2/  =  ± 

9x  rp  42/  =  6. 

2xif32/V3  +  12  =  0,  ^^   ,         «'•'      ,  ,^  ^   .         ^"^      . 

6xV3qi42/  +  5V3  =  0.          '                Va2+62  ^           V^iH^2 

r  X  +  42/  =  10,  10.   Same  answers  as  No.  9. 

3.   J  4x -2/- 6  =  0;  n.   ft2:a-2. 
(  -  8,  -  i. 

_    a2      62  „                   ft 

5.  — ^4-  — =1.  12.   X  =  ±  — 7='    2/ =  ±  — =• 

7/1-        H2  V2                           \^ 

6.  9x2  +  252/2  =  225.  13     //  =  4,  3x  +  2y  =  17. 


22  ANALYTIC    GEOMETRY. 

14-.   The  equation  ±  Vox  ±  •"?;/  =  9a  represents  the  four  tangents. 
15.   aVl  —  e-^  008^0.  16.   ^(a^csc^sec^  — c^cot^). 

17.   The  extremities  of  the  latera  recta. 

19.   The  method  of  solving  this  question  is  similar  to  that  employed 
in  §  136.     The  required  locus  is  the  auxiliary  circle  x-  +  y-  =  a^. 

Exercise  34.     Page  152. 

(a;=8,  40y  =  9x  + 72;         8.   6x  +  ay  qp  a6  V2  =  0. 

I   .1,8.      SJ2. 

_     -,,.'!..  9.   -COS0  +  T.sin^  =  1. 

2.   Within.  a  b 

ab 
10. 


3.  iV3.  ■""•    V^^ 


11.  aVl  —  e-cos2  <f>. 

12.  c2:6-'. 
2V3                                     13.   a-  — e*Xi2. 


4.   iV2. 

12.    c2:6-'. 


1  +  Vl3  _  14.   V(l-  e'^)(a2-e-^Xi2). 

6.  X  +  y  =  ±  Va2  +  52.                                   Vl  —  e'^      b 
„    1     ,  1.  /"TT"; — ;;     ,^      15-  tan  4>  = =  -• 

7.  &x  4-  cy  qi  oVa^  +  c^  =  0.  e  c 

18.   The  locus  is  the  minor  axis  produced. 

X  —     j  +  ?/2  =  f2;  centre  is  I  „,  0  j ;  semi-axes  ace 

-  and  r. 

20.   The  ellipse  o'^  (  y  —  -  j  +  ^^x^  =  — — ;  centre  is  (0,  -  j ;  semi- 

a       ,  b 
axes  are  -  and  -• 

In  21-23  take  the  base  of  the  triangle  as  the  axis  of  x,  and  the 
origin  at  its  middle  point. 

21.  The  ellipse  {.s2  —  c2)x2  +  s2?/2  =  s2(s2  —  c2). 

22.  The  ellipse  kx-  +  y~  =  kd^. 

23.  The  circle  (x  +  (")-  +  y- =  4a2. 

Exercise  35.     Page  164. 
1.    Stt.         2.    I  Vs.  3.    20x  +  63?/,— 36  =  0. 

b.    (^ ^> ^j  8.    (1)  jni2=-,  (u)mi2=-,  (m)m^2  =  l. 


ANSWERS. 


11.  3x+8y  =  4,  2x  -  3?/  =  0. 

12.  Area  =  —  (?n  +  n),    m  and  n  being  the  two  segments  (use  the 

2a 

polar  equation). 

13.  2Gx  +  33?/ -  92  =  0.  14.    x  +  2?/ =  8. 

15.    b'-z  +  a^y  =  0,  b^x -a^  =  0,  a^y  +  Wx  =  0,hx  +  ay-  0. 

17.    ahj^x  =  b^xiy.  29    ^  +  ^  =  ^. 

23.    -T  1  =  0. 

«      ^  _  a(l  -  e2) 

30.    /)  — 


24.  6xVz-  —  62  ±  a]/\/a;^—l^  =  0.  1  +  e  cos  0 

25.  e=^V6.  g^  62 

26.  See  §  148.  '    ^        1  —  e2cos2  6> 

32.  1(5x2  +  49?/2  -  128x  -  686?/  +  1873  =  0. 

33.  2a  =  18,  26  =  10.  34.    ^' +  ^2  =  5^. 

144 

c2  —  62                                               a—b 
35.    cos  0  =  \  -^ — 7:7  •  36.    tan  0  = p=- 

37.  Find  the  ratio  of  yi  to  the  intercept  on  the  axis  of  y. 

38.  h'^hx  +  aVcy  =  b^h' +  am.        ^^     „,       ,,.       x2      y2 

42.    The  ellipse  —  +  t:;  =  i. 

a2       62      - 

The  ellipse  62x2  +a2y2  =  52^2 
44. 


X2 

a2 

+ 

2/2 
62 

=  2. 

43. 

(- 

-  1 

,1) 

,  a  = 

^, 

6 

=  1, 

46.    The  ellipse  25x2  +  \C^y^-  -  48y  =  64. 


45.    A/^'A/^'inwhich£:=-F+^^%^- 


Exercise  36.     Page  174. 
s^  _  y^  _  3.    3x2  —  y2  —  ;3(i2_ 

^^      ^^  4.    G25x2  -  84?/2  =  10,000. 

4x2      ^2  _ 
^-    25~36~^-  5.    2.c2  -  22/2  =  c2. 

7.    rt  =  4,  6  =  3,  c  =  5,  e  =  |,  latus  rectum  =  |. 


24  ANALYTIC    GEOMETRY. 

8.  Idy-  —  Dx-  =  144,   transverse   axis  =  0,  conjugate  axis  =  8,  dis- 
tance between  foci  =  10,  latus  rectum  =  A/. 

9.  a:  6=1:  Vs.  11.    e  =  72.  12.    (5,-0^). 
14.   Foci,  (5,  0),  (—  5,  0);  asymptotes,  y  =  ±  |x.     17.    b. 

Exercise  37.     Page  176. 

1.    KJx  -Qy  =  28,  Ox  +  Uhj  =  100;  |,  %»-.  3.   x^  -  y^  =  0,(5,  4). 

4.    The  four  points  represented  by 

x=     ■ '  y 


Va2  -  62            Va2  -  62 
9.  — =•  10.   —^ ^  =  1. 

11.    When  a  is  less  tluin  6.  12.    Tlie  circle  x-  +  y^  =  (j2. 


Exercise  38.     Page  177. 

1.  26e,  ae2.  12.  (0,  ±  Va-'  -  62). 

2.  14  and  6  ;  (— 8,  ±  SVS).  13.  62>a2. 

3.  The  sum  =  2ex.  14.  g4x  —  Oy  —  741  =  0. 
8.  {a,  6V2),  (a,  -  6V2).  15.  y  =  ix  ±  8\^. 

10.  They  are  equal.  a'lb- 

11.  y  =  xV2  +  a.  '  a2+~62 


Exercise  39.     Page  188. 

1.    9x  +  \2y  +  10  =  0.  6.    a. 

8.  75x  -  Wy  =  0. 

9.  245x-  122/ -1189  =  0. 

10.  fVs. 

17.  See  §  140. 

18.  ^  +  ^  =  2. 

a;i     yi 


2. 

x  =  ±-- 
e 

3. 

TT 

2' 

4. 

/         >lffl2    562  X 

1.        C  '    C  j 

5. 

X  +  a  =  0. 

ANSWERS.  25 


19. 


X2        7/2^  2X_^  _  ^  5-2 


(i')    ^-K+'^  =  0. 


21. 


23.  The  hyperbola  3x2  —  y- +  20x  —  100  =  0.  The  centre  is  the 
point  ( —  L"-,  0).  Changing  the  origin  to  tlie  centre,  we  obtain  9x2 
-  3y2  =  400. 

24.  Writing  the  equation  in  tlie  form  (x  —1)2  —  4(?/  +  2)2  =  4,  and 

x2        ?/2 

changing  the  origin  to  (1,  —  2)  we  obtain  T  ~  T  —  1-     T^^  centre  i.s 
(1,  -2),  a  =  2,  6=  1. 

25.  Centre  is  ( -r-^.  TTT  h  semi-axes  are  -%/    ,   .*/— ,  in  whith 

1)2        ^2 

4J.      41? 

26.  Tlie  locus  is  the  curve  2xy  —  7x  +  4?/  =  0.  If  we  change  the 
origin  to  the  point  {h,  fc),  we  can  so  choose  tlie  values  of  h  and  k  as  to 
eliminate  the  terms  containing  x  and  y.  Making  the  change,  we  obtain 

2x1/  +  (2A;  -  7)x  +  (2ft  +  A)y  —  Ik  +  4k  +  'Ihk  -  0. 
If  we  choose  h  and  k  so  that  2A  +  4  =  0,  and  "Ik  —  7  =  0,  that  is,  if 
we  take  ft  =  —  2,  fc  =  .1,  the  terras  containing  x  and  y  vanish,  and  the 
equation  becomes  xy  =  —  7.  Hence  we  see  (§  182,  Cor.)  that  the  locus 
is  an  equilateral  hyperbola,  whose  branches  lie  in  the  second  and  fourth 
quadrants,  and  that  the  new  axes  of  coordinates  are  the  asymptotes. 

27.  The  equilateral  hyperbola  2xy  =  cfi. 

28.  Taking  the  base  as  axis  of  x,  and  the  vertex  of  the  smaller 
angle  as  origin,  the  locus  consists  of  the  axis  of  x  and  the  hyperbola 
3x2  —  ?/2  —  2ax  -  0. 

Exercise  40.     Page  206. 

1.  The  ellipse  72x2  +  48^2  =35.         8.    The  parabola  y-  =  3x  V2. 

2.  The  ellipse  4.x2  +  2//2  =  1.  9.    The  parabola  ?/2  =  2x. 

3.  The  hyperbola  32x2  —  4»y-=d.  10.    The  ellipse  4x2  +  '.)//2  =  3(5. 

4.  The  ellip.'^e  dx'^  +  ?,y-  -  32.         11.    The  point  (0,  0). 

5.  The  hyperbola 4x2  — 4//2  +  1=0.  12.    'I'ho  hyperbola  4x2  _  <),/-j  ==  ;j(] 

6.  The  parabola  y2  =  — jx.  13.  TlK'Straightlines.v=x,y=— 5. 

7.  The  parabola  y-  =  2xV2. 


26  ANALYTIC    GEOMETRY. 

Exercise  42.     Page  229. 
2.    2d  ;  5th  ;  6th  ;  7tli  ;  ;3d  ;  4th  ;  8th. 

5.  5V2;     VTi;  V83;   O.8V2,    0.4V2,    O.5V2  ;     iVli,   -  ^\y/U, 
-  xVVIi  ;  ^V83,_-  A V83,  -  /^V83. 

6.  tVVi4,  ^Vi4,  y\Vl4.     The  line  is  parallel  to  the  radius  vector 
of  the  point  (1,  2,  3). 

7.  Parallel  to  the  radius  vector  of  the  point  (A,  B,  C). 
ABC 


yjA-^  +  ii2  +  C2      V^2  4-  £2  +  c^      V^-^  +  B^+  C^ 

8.    60°  or  120°  ;  90°  ;  60°  or  120°. 

Exercise  43.     Page  235. 

1.     V3;   V3;  2V3.  3.    5V2.  4.    -  J-^y2,  -  §V2,   \S. 

5.  Lines  parallel  to  the  radius  vector  of  (3,  —  2,  —  5). 

^3^V38,   -tVV38,  -^\V38. 

6.  cos-i  ||Vl4.  9.    (1,  V3,  2V3). 

7.  90°.  12.    (t,  -  f,  I). 

8.  (4,    iTT,    iTT).  13.       (-  -2,9-,    -  -V«,    ¥)• 

Exercise  44.     Page  240. 

4.  2x  -  3?/  +  V3z  =  28. 

5.  ^-f-|=l,T  +  f  +  ^=-l,2x-|2/  +  62  =  0. 
4       .5       /  1       z       5 

6.  6x  +  2/  —  z  =  5. 

7.  cos-i  ^-.  12.    -  +  ^  -  -  =  1  ; 

8.  79°  52'.  ■    a      6      c 

(J  —  2abc ^ 

^'   '^^^"^  Vl^TW^C^'  V6%Mr^2c2  +  a262 

^  14.    2x  4-  4?/  —  2  =  23. 
COS — ^  —           —              -'  1 

V^2  +  ^2  +  (72  16.    3x  —  ?/  -f  2z  =  4. 

i?  18.   .c  +  y  +  z  =  0 ;  2V3. 

cos-'i     , 

V.42  +  i?2  +  C2  19.    ?,x  -  iy  +  7z  +  13  =  0. 

10.    2.25.  20.    2x  +  5?/  —  z  =  9. 


ANSWERS.  27 

Exercise  45.     Page  247. 

3.   It  passes  through  (i,  0,  —  |),  and  is  parallel  to  the  radius  vector 
of  (4,  5,  3). 

x^zA^lL^zl^^^il.  6.    (4,  5,0),  (0,  1,4),  (-1,0,  5). 

2              2            —  2  '  7.    22/  +  5z  =  10. 

8.   2x  —  5?/  +  1  =  0,  2x  +  5z  +  1  =  0,  ?/  +  z  =  0. 

11.  cos -1—0.1  x  —  'P._y+7_z  +  o 

12.  14° 57' 45".  '     -3   ~      5      ~   -  6 

r-  ,„x  —  2w  +  4z  +  6 

15.   ?/  =  -  2x  -  1,  z  =  3a;  +  5.     18.  sin-i  ^%. 
X  —  p        _        y  —  Q        _  z 


19. 


1 


Vl  +   »l2  +  ?l-  Vl   +  7)1-  +  n-  Vl  +  H(2  +  ,j2 

in  which  the  denominators  equal  cos  a,  cos  p,  cos  7,  respectively. 

Exercise  46.     Page  263. 

1.  The  planes  x=—  4,  x  =  —  1,  x  =  2;  the  planes  y  =  —  2,  y  =1, 
y  =  3  ;  the  planes  z  =  0,  z—  —  m. 

2.  Answer  to  the  first,  the  parabolic  cylinder  whose  elements  are 
parallel  to  the  axis  of  z,  and  wliose  trace  on  the  plane  xy  is  the  parab- 
ola 2/2  =  8x. 

3.  5x2  +  10^2  -  5(5^  5^2  _  j^2  =  8,  x2  +  13z2  =  32. 

4.  X-  +  ]/-=  i,  z  =  i:  |V42  ;  hence  the  curves  are  two  circles  in 
the  planes  z_—  ±  sV42,  whose  centres  are  in  the  axis  of  z,  and  whose 
radii  are  Vf-  each. 

5.  a  =  3,  6  =  2,  e  =  iV5.  _ 

6.  Two  circles  whose  radii  are  •\/l3  each. 

8.  9(x2  +  y"-)  =  (z  -  5)2  ;  x2  +  2/2  =  2^5. 

9.  25x2  +  252/2  -  9z2  +  90z  =  225. 

10.  x2  +  2/2  —  |z  —  I  =  0. 

11.  2/2  +  22  -  8x  =  0'. 

12.  x2  +  y2  _  J22  -  3z  =  9 ;   (0,  0,  -  6). 

13.  9(x2  +  2/2)  +  4z2  =  36. 

14.  lG(x2  +  y-)  +  9z2  =  144. 

15.  4(x2  +  2/2)  _  9^2  =  3(5. 

16.  x2z*  +  y-z*  =  1,  xf  +  2/2  -  iz3  =  0. 

17.  a  =  5V3,  b=  jV6. 


WENTWORTH'S||f^° 
TRIGONOMETRIES  ^pitiqns 


THE  aim  has  been  to  furnish  in  these  books  just  so  much  of 
trigonometry  as  is  actually  taught  in  our  best  schools  and 
colleges.  The  principles  have  been  unfolded  with  the  utmost 
brevity  consistent  with  simplicity  and  clearness,  and  interesting 
problems  have  been  selected  with  a  view  to  awakening  a  real 
love  for  the  study.  The  subjects  of  Surveying  and  Navigation 
are  presented  in  accordance  with  the  best  methods  in  actual  use, 
and  in  so  small  a  compass  that  the  general  student  may  find  time 

to  acquire  a  competent  knowledge  of  them. 

List  price 
Plane  and  Solid  Geometry  and  Plane  Trigonometry.      i2mo. 

Half  morocco.     655  pages 51.40 

Plane  Trigonometry.     lamn.     Cloth.     162  pages 60 

Plane  Trigonometry,  and  Tables.    Svo.    Cloth.    258  pages     .     .       .90 
Plane  and  Spherical  Trigonometry.    i2mo.    Half  morocco.    232 

pages 85 

Plane  and  Spherical   Trigonometry,  and  Tables.     Svo.     Half 

morocco.     32S  pages 1.20 

Plane  Trigonometry,    Surveying,   and    Tables.       Svo.      Half 

morocco.     357  pages 1.20 

Surveying  and  Tables.     Svo.    Cloth.     194  pages So 

Plane  and  Spherical  Trigonometry,   Surveying,  and  Tables. 

Svo.     Half  morocco.     427  pages 1.35 

Plane  and  Spherical  Trigonometry,  Surveying,  and  Navigation. 

i2mo.     Half  morocco.     412  pages 1.20 

Logarithms,  Metric  Measures,  and  Special  Subjects  in  Advanced 

Algebra.     i2mo.     Paper.     141  pages 25 


New  Five-Place  Logarithmic  and  Trigonometric  Tables.     By 

G.  .A.  Wentworth  and  (1.  .\.  Hill 

Seven  Tables   (for    Trigonometry   and  Surveying).     Cloth. 

7<)  pages 50 

Complete  (for   Trigonometry,    Surveying,  and    Navigation). 

Half  morocco.     154  pages i.oo 


GINN    6    COMPANY    Publishers 

BOSTON  NEW  YORK       CHICAGO       LONDON 

SAN  FRANCISCO    ATLANTA       DALLAS       COLUMBUS 


fVENTfVORTH'S 

COLLEGE   JILGE'BRA 

REVISED    EDITION 

i2mo.    Half  morocco.    530  pages.    List  price,  $1.50;  mailing  price,  f  1.65. 


This  book  is  a  thorough  revision  of  the  author's  "College 
Algebra."  Some  chapters  of  the  old  edition  have  been 
wholly  rewritten,  and  the  other  chapters  have  been  rewritten 
in  part  and  greatly  improved.  The  order  of  topics  has  been 
changed  to  a  certain  extent ;  the  plan  is  to  have  each  chapter 
as  complete  in  itself  as  possible,  so  that  the  teacher  may  vary 
the  order  of  succession  at  his  discretion. 

As  the  name  implies,  the  work  is  intended  for  colleges  and 
scientific  schools.  The  first  part  is  simply  a  review  of  the 
principles  of  Algebra  preceding  Quadratic  Equations,  with 
just  enough  examples  to  illustrate  and  enforce  these  princi- 
ples. By  this  brief  treatment  of  the  first  chapters  sufficient 
space  is  allowed,  without  making  the  book  cumbersome,  for 
a  full  discussion  of  Quadratic  Equations,  The  Binomial 
Theorem,  Choice,  Chance,  Series,  Determinants,  and  The 
General  Properties  of  Equations. 

Every  effort  has  been  made  to  present  in  the  clearest  light 
each  subject  discussed,  and  to  give  in  matter  and  methods 
the  best  training  in  algebraic  analysis  at  present  attainable. 

ADDITIONAL     PUBLICATIONS 

By  G.  A.  WENTWORTH  List  Mailing 

price  price 

Plane  and  Solid  Geometry  (Revised  Edition)    .     .     .  ^1.25  $1.40 

Plane  Geometry  (Revised  Edition) .75  .85 

Solid  Geometry  (Revised  Edition) .75  .85 

Analytic  Geometry 1.25  1.35 

Trigonometries  (Second  Revised  Editions)  .... 

A  list  0/  the  various  editions  of  Wentiuorth's  Trigonometries  will  be 
sent  on  request. 

GINN    &    COMPANY    Publishers 


SCIENCE   AND  ENGINEERING 

LIBRARY 

University  of  California, 

San  Diego 

'                                  DATR    nilF. 

;  0CT2  8  1S75 

'   ^ JDSr  1 4  mo 

MAY  0  "^  1QRfi 

ii\r\{   V  <o  IvJOO 

MAY  2  8 1986 

mN2  4l986 

JUL  2  9 1986 

nnrn  11986 

, 

i 

''       SE  16 

VCSU  Lihr. 

UC  SOUTHERN  REGIONAL  LIBRARY  FACILITY 


A  A  001  413  993  5 


^. 


